DOCUMENT RESUME ED 391 849 UD 030 791 AUTHOR TITLE 95 … · DOCUMENT RESUME. ED 391 849. UD 030 791. AUTHOR Rogers, Pat, Ed.; Kaiser, Gabriele, Ed. TITLE Equity in Mathematics Education

DOCUMENT RESUME

ED 391 849 UD 030 791

AUTHOR Rogers, Pat, Ed.; Kaiser, Gabriele, Ed.TITLE Equity in Mathematics Education. Influences of

Feminism and Culture.REPORT NO ISBN-0-7507-0401-2PUB DATE 95

NOTE 287p.; Many chapters derived from sessions on "Genderand Mathematics Education" at the meeting of theInternational Congress on Mathematical Education(7th, Quebec.City, Quebec, Canada, August 1992).

AVAILABLE FROM Falmer Press, Taylor & Francis Inc., 1900 Frost Road,Suite 101, Bristol, PA 19007 (paperback:ISBN-0-7507-0401-2, $24.95; clothbound:ISBN-0-7507-0400-4).

PUB TYPE Books (010) Collected Works General (020)

Reports Evaluative/Feasibility (142)

EDRS PRICE MF01/PC12 Plus Postage.DESCRIPTORS Curriculum Development; *Educational Change;

Educational Practices; *Equal Education; *Females;*Feminism; *Mathematics Education; MathematicsInstruction; Models; Sex Differences; *Sex Fairness;Teaching Methods

ABSTRACTThis book provides educators and other interested

readers with an overview of the most recent developments and changesin the field of gender and mathematics. The overview is grounded in amodel for understanding how chaAge occurs. The model, developed by P.McIntosh (1983), arose from.the examination of efforts in NorthAmerica to liberate mathematics from a maLe-dominated Eurocentricworld view and to develop a more inclusive curriculum. Anintroductory chapter describes the McIntosh model, which moves from"womanless mathematics" through stages to a reconstructedmathematics. Twenty-six additional chapters are.grouped into thefollowing sections: (1) "Intervening with Female Students"; (2)

"Working with Female Teachers"; (3) "Focusing on PracticingTeachers"; (4) "Educating the Public"; (5) "Comparative Studies"; (6)

"Cultural Perspectives"; (7) "Feminist Pedagogy in MathematicsEducation"; and (8) "Changing the Discipline." References follow eachchapter. (Contains 1 figure and 22 tables.) (SLD)

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Equity inMathematics Education

Tb our damhters

Kate (ind Sarah

Aj

Equity inMathematics Education:

Influences of Feminismand Culture

Edited by

Pat Rogers and Gabriele Kaiser

The Falmer Press

(A member of the Taylor & Francis Group)London Washington D.C.

Li

UK The Falmer Press, 4 John Street, London WC IN 2ETUSA The Falmer Press, Taylor & Francis Inc.. 1900 Frost Road, Suite 101,

Bristol, PA 19007

P. Rogers and G. Kaiser. 1995

.411rights reserivd. No part i!f this publication 'nay be ieprodived, stored in a retrievalsystem, or transmitted in any j.011,1 or by any meaw, electronic, mechankal.photocopying, recording or otnern.ise, without permission in writing.from the Publisher.

First published ni 1995

A catalogue record for this book is available from the British Library

Library of Congress Cataloging-in-Publication Data are availableon request

ISBN 0 7507 0400 4 (Cased)ISBN 0 7507 0401 2 (Paperback)

Jacket design by Caroline Archer

Typeset in 10/12pt Bembo bySolidus (Bristol) Limited

Prhited in Great Britain by Burgess Science Press. Basingstoke on paper which has aspecified pH inlite on Jima paper manufacture of not less than 7.5 and is thercfore 'acid

0

Contents

Acknowledvmms

Chapter 1 Introduction: Equity in Mathematics EducationGabriele Kaiser and Pat Rogers

Part 1 Effecting Change 11

Intervening with Female Students

Chapter 2 Connecting Women with MathematicsCharlene Morrow and _James Morrow

Chapter 3 The METRO Achievement Program: HelpingInner-City Girls ExcelDenisse R. Thompson

Working with Female Teachers

Chapter 4

Chapter 5

13

17

Who Wants to Feel Stupid All of the Time?Olive Fullerton 37

Assisting Women to Complete Graduate DegreesLynn Friedman 49

Focusing on Practising Teachers

Chapter 6

Chapter 7

Chapter 8

A National Network of Women: Why, How, and ForWhat?Barbro Grevholm 59

Women Do MathsClaudic Solar, Louiq. 1.4ornuw and HAw Kayler 66

'Girls and Computers': Making Teachers Aware of theProblemsCornelia Niederdrenk-FeIgner 72

Contents

Educating the Public

Chapter 9

Part 2

Common Threads: Perceptions of Mathematics Ed. anionand the Traditional Work of WomenMary Harris 77

The Cultural Context 89

Comparative Studies

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Culture, Gender and MathematicsGurtharn Singh Kaeley 91

Why Hawai'i Girls Outperform Hawaii Boys: TheInfluence of Immigration and Peer CulturePaul Brandon, CathieJordan and 'filly Ann Higa 98

Mathematics and New Zealand Maori GirlsSharleen D. Forbes 109

Equal or Different? Cultural Influences on LearningMathematicsGilah Leder 117

Cultural Perspectives

Chapter 14 Gender Inequity in Education: A Non-WesternPerspectiveSaleha Nagluni Habibullah 126

Chapter 15 Gender and Mathematics: The Singapore PerspectiveBerinderjeet Kato- 129

Chapter 16 Gender and Mathematics Education in Papua NewGuineaNeela Sukthankar 135

Chapter 17 The French Experience: The Effects of De-SegregationFrancoise Delon 141

Chapter 18 The Contribution of Girls-Only Schools to Mathematicsand Science Education in MalawiPat Hiddlestort 147

Part 3 Feminist Pedagogy in Mathematics Education 153

Feminist Influences

Chapter 19 Feminism and Strategies for Redressing GenderImbalance in MathematicsRoberta Mura 155

ContentS

Chapter 20 Women's Ways of Knowing in MathematicsJoanne Rossi Becker 163

Chapter 21 Putting Theory into PracticePat Rogers 175

Chapter 22 Gender Reform through School MathematicsSue Willis 186

Chapter 23 We Don't Even Want 'to Play: Classroom Strategies andCurriculum which Benefit Girls_Patina Hivins 200

Changing the Discipline

Chapter 24

Chapter 25

Chapter 26

Moving Towards a Feminist Epistemology of MathematicsLeone Burton 209

Mathematics: An Abstracted DiscourseBetty.fohtIstoti 116

Constraints on Girls' Actions in Mathematics Educationfarjolijn Il'itt 135

Epilogue 145

Chapter 27 Mathematics: Beyond Good and Evil?Nalicy Shelley

Notes on Contributorshidex

247

263271

vii

Acknowledgments

We would like to acknowledge the excellent work of Steven Katz who assisted uswith the editing of this book. His careful checking of references and hispainstaking attention to the detail and form of individual chapters were invaluableto us and to the authors.

We wish to thank Kluwer Academic Publishers to reproduce Chapter 24 ofthis book which appeared first in Edumtional Studies in Matlu7naties, 28, 3, 1995.

ix

Chapter 1

Introduction: Equity in MathematicsEducation

Gabriele Kaiser and Pat Rogers

Research and intervention over the past three decades have greatly increased ourunderstand'Ag of the relationship between gender and participation in mathe-matics education. Research, most of it quantitative, has taught us that genderdifferences in mathematics achievement and participation are not due to biology,but to complex interactions among social and cultural factors, societal expecta-nom, personal belief systems and confidence levels. Intervention to alter theimpact of these interactions has proved successful, at least in the short term.Typically, interventions sought to remedy perceived 'deficits' in women's attitudesand/or aptitudes in mathematics by means of 'special programmes' and 'experi-mental treatments'. But recent advances in scholarship regarding the teaching andlearning of mathematics have brought new insights. Current research, profoundlyinfluenced by feminist thought and methods of enquiry, has established how afuller understanding of the nature of mathematics as a discipline, and different,more inclusive instructional practices can remove traditional obstacles that havethwarted the success of women in this important field. Some argue that practicesarising out of contemporary analysis will improve the study of mathemancs for allstudents, male and female alike.

This book provides teachers, educators and other interested readers with anoverview of the most recent developments and changes in the field of gender andmathematics education. Many of the chapters in this volume arose out of sessionson 'Gender and Mathematics Education' organized by the editors for IOWME(International Organization for Women and Mathematics Education) as part ofICME-7 (the Seventh International Congress on Mathematical Education) held inQuebec City, Canada in August 1992. We are fortunate to have in one volume theperspectives of internationally renowned researchers and practitioners from all overthe worldind from a variety of ethnic and cultural backgrounds.

However, the book does more than provide a review of current thinking inthe area for we have grounded our overview in a model for understanding howchange occurs. This nmdel, developed by Peggy McIntosh (1983), arose out of anexamination of the evolution of effbrts in North America to loosen curriculum

1 1.


from a male-dominated, Eurocentric world view and to evolve a more inclusive

curriculum to which all may have access. We acknowledge the danger and thedifficulties inherent in attempting to classify anything into what might be narrowor restrictive cacegories. Nonetheless, we believe that applying this model tomathematics education provides new insight and guidance for future endeavours.Using the model as a lens for examining attempts to change the relationshipbetween gender and mathematics education, we are able to discern and questionunderlying assumptions, to appreciate where we have been and to understand howcertain feminist theories and cultural influences have affected and transformed ourpractical efforts to implement changes suggested by the research. We are then ableto ask the important questions: where are we heading and to what end?

In this introductory chapter, we describe the McIntosh model and locatework in the area of gender reform of mathematics education in phases of hermodel (for an earlier application of the McIntosh nmdel to mathematicseducation, see Countryman, 1992). In terms of the McIntosh model, we are in atransitional stage, between Phase Three (seeing women as victims or as problemsin mathematics) and Phase Four (seeing women as central to the development ofmathematics). This book is organized to lead the reader through this transition; aswell it presents some critical perspectives and insight from researchers indeveloping countries which broaden and enrich the discussion.

Understanding Curriculum Reform

According to Peggy McIntosh, her typology of interactive phases of personal andcurricular revision derive from her work in helping `traditionally trained whitefaculty meml rs to bring into the liberal arts curriculum new materials andperspectives from women's studies' (McIntosh, 1989). The model comprises fivestages of awareness which, according to McIntosh, are patterns of realization orframes of mind which occur in succession as individual scholars re-examine theassumptions and grounding of their discipline and enlarge their understanding ofthe field. For example, in the field of history,

[t]raditionally trained white ... [historians] ... were likely to move fromthinking and teaching in Phase One: Womanless History, to Phase Two:Women in History. Then followed an expansion into Women as aProblem, Anomaly, Absence or Victim in History. All of these are, or canbe, conceptually male-centred. I identify as Phase Four the far moredaring Women's Lives As History, and looked toward Phase Five: HistoryRedefined and Reconstructed to Include Us AIL (McIntosh, 1989)

The issue for the authors of the chapters in this book is how to reformmathematics education to *include us all', or more specifically, to include a greaterproportion of women. Effbrts in the west to achiew gender balance inmathematics education have derived from the essentially monocultural view thata certain measure.of familiarity and competence with mathematics is important to

2

Introduaion

every individuals future growth and economical survival. According to McIntosh(1989), those who 'think monoculturally about others, often imagine ... thatothers lives must be constituted of "issues" or "problems" . Thus, inattempting to change a perceived gender imbalance in mathematics participation,we have tended to see women who do not embrace mathematics as deprived, ashaving 'a problem' which we need to fix. When we enlarge the scope of ourreform efforts, another danger of monocultural thinking is evident when weblindly apply methods that have been s:accessful in one culture or one ethnic groupto another without any, or at best with only minimal, adaptation to local needs andcircumstances (see in particular, Chapter 14 by Saleha Naghmi Habibullah, for aspirited attack of this view).

Applying McIntosh's model to mathematics, we discern five phases, which wename (adapted from Countryman, 1992, p. 84):

Phase One: Woman less mathematics;Phase Two: Women in mathematics;Phase Three: Women as a problem in mathematics;Phase Four: Women as central to mathematics; andPhase Five: Mathematics reconstructed.

Thi5 book is organized according to these phases. In this introductory chapterwe draw extensively on McIntosh (1989) to describe the first three (essentiallymonocultural) phases of the model in the context of reform in mathematicseducation. However, we should warn the reader, as does McIntosh herself, thatthese phases do not always occur in the succession given here. Individuals mayweave back and forth between and among the phases. Part 1 provides an overviewof approaches to gender reform of mathematics education which challenge themonocultural assumptions and deficit philosophy of Phase Three and by virtue oftheir debt to feminist influences exhibit qualities which place them somewhere inthe trmisition between Phase Three and Four. It is also important to remember thatthe McIntosh model has grown out of a North American perspective. Thesituation may be very different in other countries zaid in particular in thedeveloping world, an issue which is raised in Part 2 of the book. Furthermore, notall western countries are currently at the same stage of awareness of the issues. Fore \ample, scholars in Germany and Sweden began to work in this area only in thelate 1980s (see, for example, Kaiser-Messmer, 1994 and Grevhohn, this volume,('hapter 6), while work in England, Australia and North America has been inprogress for over two de:ades. The chapters in Part 3 describe recent advances indeveloping a Phase Four approach to mathematics education, one in whicalwomen's experience is central to the discipline and to its pedagogy. The chapterwe have used as our epilogue allows us to imagine where we will be when ourwork is done.

t),3

Gabriel( Kaiser and Pat Rogers

The Early Phases of Gender Reform in Mathematics Education

Phase One: Womenless Mathematks

When the editors of this book were in secondary school, it was unusual for women

to specialize in mathematics beyond the compulsory years of schooling, even rarerfor them to go on to university to study mathematics. None of the theorems listedin our mathematics textbooks were named after women, or if they were, this fact

was not made apparent to us. The language of instruction was unselfconsciouslymale, examples dealt with male experience. One of us, Rogers, developed a habitof personifying mathematical objects and was not even aware that she did so until

years later when one of her students pointed out that she referred to mathematicalterms and objects exclusively as 'him' or 'he'. Before leaving school to enrol in aprestigious university to study mathematics, she did not think it incongruous that

the book prize she received for excellence in mathematics was entitled Men qfMathematks (Bell, 1937). Mathematics was what men did. We learned that womenhad not been necessary to the development of mathematics, and were unlikely to

be essential to its future development. In this phase, many women whonevertheless pursue mathematics experience silence and exclusion, a feeling that

'this is not about me', a feeling not unlike the first stage of knowing described byBelenky et al. (1986) and discussed by Becker in Chapter 20.

Phase livo: Ptimien in Mathematics

This phase in mathematics education reform began in North America in the 1970swhen scholars began to investigate the lives and works of the few exceptionalwomen throughout history who had been successful in mathematics, for example,Hypatia, Sonya Kovalevskya, and Emmy Noether. Although this phase challenges

the all-male face of success in mathematics, it presents no challenge to the termsof success. Despite its roots in early North African civilizations, 'Mathematicsremains what white men do, along with a few token women' (Countryman, 1992,p. 77). The history of exceptional women in mathematics was injected into ourexperience (see, for example, Lynn Osen's 1974 reply to E.T. Bell, Women (fALithemath-s). The problem with the Phase Two 'famous few' approach(McIntosh, 1989) ;. that it teaches about women mathematicians as exceptions and

can convey the impression that most women do nothing of any value inmathematics, and that they can only be worthy of ootice if they become more likemen. It ascribes to women in the field a 'loner' status that makes them vulnerableto every setback. It leaves women mathematicians believing that they can make iton their own by virtue of sheer hard work, and that if they don't, their failure hassomething to do with personal merit rather than with the way mathematics cultureis organized. In contrast, those who are successful may begin to fear they will notbe seen as female anymore. For example, Rogers, who went to an all-girlssecondary school, learned to be proud of being good at something in whichwomen didn't excel she was one of only three girls who continued to study

4

Introduction

mathematics to the end of secondary school and the only one who ultimatelypursued a career in pure mathematics. But privately, she also began to experiencea nagging feeling that perhaps she wasn't a real women. Her self-image during thisperiod of her life was that of a long bean-pole with a large pulsating sphere on thetop. Her internal struggle between the life of a woman and the life of the mindmade it difficult for her to embrace the intellectual life she enjoyed and retain anintegrated sense of herself at the same time.

A further problem with this 'woman mathematician as exception' phase is thatit focuses on individual success, on 'winners and losers', and thus devalues thosewho prefer collaborative approaches. Group work is thus not encouraged in theclassroom and mathematics is seen as the property of experts. Again referring toBelenky et al., this is a phase that encourages and promotes a received view ofknowledge (see Becker, Chapter 20).

Phase Three: Iii)men as a Problem in Mathematics

An illustration of a shift in approach to Phase Three is provided by the residentialmathematics camps for 15 year-old girls, entitled 'Real Women Don't Do Math!'(Rogers, 1985), which Rogers organized in the summers of 1985 and 1986. Thetitle of this camp was inspired by di.' phrase 'real men don't eat quiche'. popularlyused at that time to challenge conventional notions of what it meant to be a man.In the playful title of Rogers' camp, we can discern the shift from seeing thewoman mathematician as a 'loner' to viewing mathematics as for all women. Wecan also sense the frustration with the prevalent assumption that mathematics is dfield in which women have difficulty (this is perhaps more evident in the title ofthe video that grew out of the camp: 'Real Women Don't Do Math! Or DoThey?').

In North America since the early 1980s, and in other English-speakingwestern countries such as Britain and Australia, gender reform of mathematics hasbeen dominated by intervention projects. The largest and most influential of theseprojects is the North American EQUALS project which has developed a varietyof teaching materials, workshops and one-day careers conferences (with titles like'Expanding Your Horizons') with the explicit purpose of increasing women'sinterest in mathematics. The aim was to make female students more aware of theneed for mathematics in an increasingly technologically driven world and of theimportance of keeping their career options open in order to compete successfullyin it, and to make their teachers and parents aware of the problem of women's poorparticipation in mathematics (see, for example, Kaseberg et al., 1980).

Undoubtedly programmes such as these have had a crucial influence on thees olution of our thinking on the issue and they have met with much success inachieving their goals. Hqwever. as mentioned earlier, a difficulty with theseapproaches is that they derive from a nionocultural perspective that does not noticethat women may have made conscious choices to avoid a subject that was in itselfalienating. Rather, it is assumed that women avoid mathematics because ofignorame concerning the importance of mathematics to their futures and the dire

5


consequences should they avoid mathematics. A different, still popular means ofintervening, is to focus on inath anxiety, a term invented by Sheila Tobias (1978,1994) to describe a psychological fear or anxiety associated with engaging inmathematical activity This approach uses essentially clinical means to help womenovercome their anxiety towards, and hence their avoidance of, mathematics. It too,although for different reasons, places the blame for lack of participation inmathematics firmly on the shoulders of women themselves. In all of theseapproaches, the mathematics itself is not questioned, only the learners. It is

assumed that women have to come to terms with their problems (their ignoranceof consequences. their faulty beliefs, their 'mathophobia'), and that when they do

all will be well.The views of proponents of such intervention programmes are typical of

liberal feminism (see Leder, Chapter 13), or feminism of .equality (see Mura.Chapter 19), in that they "work within the system,- attempting only to improvethe lot of women within a society which is otherwise left unchanged' (Damarin,1994). It is evident in this phase, that while the focus is on important issues, suchas sexism and oppression, in attempting to make mathematics a more opendiscipline the emphasis is on disempowerment (why women can't do mathematics,why they avoid it) rather than on empowerment (how women can learn the skillsto challenge the discipline and change the mathematics). Programmes ofremediation, self-help and career information do not fundamentally challenF thepower of authorities. In fact they may even serve to maintain the status quo.

Gender and Mathematics Education: The State of the Art

The first part of this book provides a general overview of the approaches that havebeen developed during the last decade to increase women's participation inmathematics. Because of their focus on changing women, there is a temptation toidentify these approaches with Phase Three in the McIntosh model. Yet, they haveall been influenced by an awareness of the systemic reasons for women's under-representation in mathematical fields and for this reason, we ascribe them atransitional status between Phase Three and Phase Four (see below).

A liansitional Phase

The chapters in Part 1 of this collection exhibit a worldwide shift of the debateon the relationship between mathematics and gender, one that is profoundlyinfluenced by feminist scholarship and by research on the cultural dependency ofapproaches. and indeed of performance and participation in mathematics itself.This can perhaps be seen most clearly in the evolution of the SummerMathProgram described by Charlene and James Morrow This summer camp for girlswas originally conceived as an intervention programme seeking to encourage girls

to do more mathematics. Over the years. however, influencedby feminist theory,in particular Women's 114s of Knowiq (Belenky et al., 1986), and by constructivist

6

1 0

Introduction

approaches to the learning of mathematics, the programme has changed itstheoretical orientation. Now in this camp there is an increased emphasis onencouraging and supporting connected knowing and different ways of teachingmathematics which conform to women's preferred learning styles.

Part 2 provides a cultural perspective on the approaches in Part 1. Thesestudies question the assumption that everyone must achieve a certain level ofmathematical competence, and raise doubts as to whether models developed inwestern countries to achieve gender equity will be effective in all countries, oreven, for that matter, with powerless groups within western countries. They pointrather to the necessity of developing individual cultural-specific approaches. Forexample, Sharleen Forbes, in analysing the situation of Maori girls in NewZealand, reports that strategies which have been successful with girls of Europeanorigin have had no positive impact on indigenous populations. She argues insteadfor strategies based on an intimate knowledge of Maori culture. Other contribu-tions in this part of the book are influenced by the culture dependence of genderdifferences. For example, Paul Brandon, Cathie Jordan and Terry Ann Higa, citingsocio-cultural causes, point out that girls from certain immigrant groups inHawafi achieve significantly better levels of mathematical attainment than boys.The chapters in this part of the book illuminate the results of SIMS (SecondInternational Mathematics Study), which show in particular that differences inmathematical attainment between countries ace larger than those between thesexes (see, for example, Hanna, 1989). The chapters by Francoise Delon andPat Hiddleston, among others, reflect another orientation within this part of thebook: the effects of single-sex education. They conic to the conclusion thatsex-segregated education has been crucial in promoting girls' mathematicsachiek ement.

The spirit of the transition from Phase Three and the promise of Phase Fouris captured by the following quote from David Henderson, a white USmathematician:

Recently, I was thinking back over the times that my perception ofmathematics had been changed by the insights or questioning of a personin [one of my courses]. Suddenly, I realized that in almost all of those casesthe other person was a woman or from a different culture than myown.... [W]hen I listen to how other people view mathematics myunderstanding of mathematics changes. I am certain that as women, andmembers of the working class and other cultures, participate more andmore in the established mathematics, our societal conceptions ofmathematics will change and our ways of perceiving our universe willexpand. This will be liberating to us all. (Henderson, 1981, p. 13)

In the next section, we describe the fourth phase in the McIntosh model andsuggest how the contributions in Part 3 of this book elucidate the process ofchange in this phase.

i 7

Gabriel(' Kaiser and Pat Rogers

Phase Four: Wonum as Central to Mathematics

this phase, women's experience and women's pursuits are made central to thedevelopment of mathematics. Work in this phase seeks to uncover privilege andto redistribute power. It emphasizes cooperation over winning and losing,difference and multiplicity over 'one right way'. Fundamentally, Phase Four shifts

the 'blame' away from women by seeking to change the system, not the women.The chapters in Part 3 anticipate this phase and are grouped into two

categories according to where they place their primary emphasis: changing thepedagogy or changing the discipline. Influenced by feminism and feministpedagogy, the chapters of the first group call for a fundamental change inpedagogical processes, one that considers the experiences of women as central totheir mathematics development, and in which emotion and reason play balancedroles. Fundamental to these approaches is the theory developed by Gilligan (1982)which describes the 'different voice' of women, so frequently silenced in academia.Influential, too, is the approach to teaching developed by Belenky et al. (1986) in

I-Vays of Know*, based on their categorization of the stages in whichwomen come to know Several chapters (see, for example, Joanne Rossi Beckerand Pat Rogers) describe how to apply feminist pedagogy to mathematics

teaching.The approaches in the second group all question the discipline of mathematics

but depart from a variety of positions. Referring to the feminist critique of thenature of science, Leone Burton's focus is developing a new epistemology ofmathematics, one which does justice to women. Betty Johnston reflects on theinteraction between mathematics and society, and especially on how thequantification of society both constructs and regulates our lives as women.Marjolijn Witte proposes a view of mathematics, based on constructivism, whichmight be more conducive to women's learning preferences.

Within the approaches covered in this part of the book, it is evident howmuch the discussion of gender equity in mathematics education has changed in thelast decade. As already mentioned at the beginning of this chapter, Pat Rogersstarted her work in the area of gender and mathematics with the organization ofmathematics camps for girls. At this time, it was widely believed that if we changedthe conditions in which students learned, and the nature of their experience withmathematics, then the relation of girls to mathematics would automaticallyimprove. During the development of her work in this area, it became clear toRogers that mere system-tinkering was not adequate to the problem at hand, andthis is reflected in the subtitle of her (1985) paper, 'Real women don't do math

With good reason!' . Since then, she has developed an approach thatfundamentally changed her own mathematics teaching, her relationship to herstudents, and the learning experience she provides. A similar shift can be observedwithin the change of the position of Leone Burton from her approach in the 1980s(see Burton, 1986) to her questioning of mathematics epistemology in this book.

In this context, it is also interesting to note that alongside debates aboutspecifically female ways of knowing, fundamental changes in research method-

8

Introduction

ology have also taken place. Most of the studies that were conducted during thePhase Three period of reform were designed to describe differences between boysand girls in achievement, or differences in their attitudes towards mathematics.Generally speaking, they were based on large student samples and the results wereusually analysed using traditional quantitative methods. Representative of this kindof research is Armstrong's analyses as part of the NAEP (National Assessment ofEducation Progress) (Armstrong, 1981, 1985). In the course of developing anemphasis on women's ways of knowing and connected teaching or feministpedagogy, qualitative methods, which have their root in case studies, have becomemore and more important. Elizabeth Fennema (1994) in asserting the importanceof continuing to sustain 'sonic research which utilizes a positivist perspective',further stated, 'I also believe that an understanding of gender and mathematicsderived [only] from studies done from [a positivist] perspcoctive will be limited. Wewill not deepen our understanding of gender and machematics until scholarlyefforts conducted in a positivist framework are complemenzed with scholarlyefforts that utilize other perspectives [for example, from cognitive science andfeminism], many of which are currently being utilized in mainstream education'.

Conclusion: Mathematics Reconstructed

In conclusion, we ask what might mathematics be when it is reconstructed toinclude us all? In Phase Five (Mathematics reconstructed), cooperation andcompetitiveness jre in balance and mathematics will be what people do. McIntoshsees 'the work towards Phase Five taking one hundred years because it involves areconstruction of consciousness, perception and behaviour.' For Countryman(1992), 'The words to describe the fifth phase are more elusive. I find it hard tosay what the transformed mathematics curriculum will be, or how we will achieveit. Surely in this phase mathematics will help us see ourselves and others inconnection with the world', (p. 75). Surely too, it will involve a fundamental shiftin what we value in mathematics, in how we teach it, in how mathematics is used,and in the relationship of mathematics to the world around us.

We have used the words of Nancy Shelley to describe the final phase inmathematics education reform. In the epilogue, she questions the disciplines ofmathematics and mathematics education. Departing from a traditional academicstyle of writing, her form itself as much a part of her argument as the argumentsshe sunmions, she stimulates us to imagine a mathematics not dominated byauthorities. Shelley touches on every issue raised in this book. In questioningmonocultural views of mathematics and mathematics education, and in examiningthe epistemological status of fundamental issues of mathematics and mathematicsteaching, she develops a vision of another type of mathematics. We leave it to ourreaders to say whether this is a vision of a mathematics close to that Henderson(above) described as 'liberating to us all'.

9

Gahriele Kaiser and Pat Rogers

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New Orleans, April 1994.Gn I I(1AN, C. (1982) In a Difkrent lice. Cambridge. MA, Harvard University Press.HANNA, G. (1989) 'Mathematics achievement of girls and boys in grade eight: Results from

twenty countries', Educational Studies, 20, pp. 225-32.

HENI /111.soN, a (1981) 'Three papers', Fot the L..earning of Mathemarks, 1,3. pp. 12-15.

KAISEIL-MESsMER. G. (1994) 'Analysen "Frauen und Mathematik- Nachbetrachtun-

gen' , Zentralhlatt.fiir Didaktik der Alathematik, 26, pp. 63-5.KANE111111,(1. A., KREIN111.1w. N. and Dowmir, D. (1980) 'Se EQUALS to Promote the

Participation of Muter: in Afar/marks, Berkeley CA, University of California.

McIN iosi i, P. (1983) Phase Theory of Curriculum Rclorm, Wellesley. MA. Center for

Research on Women.Mc IN HP,11, P. (1989) Interactive Phases of Curricular and Personal Re-vision 117th Re,ord to

Race, Wellesley MA. Center for Research on Women.Ost N, TM. (1974) FT imien in Matlumatirs,Cambridge, MA, MIT Press.12.0( ;Elks, P. (1985) 'Overcoming another barrier: Real women don't do math With

good reasonr. Canadian liilmatt Studio Journal, 6.4 (Winter), pp. 82-4.

Toums, S. (1978,1994) Ovoroming Math Anxiety, New York, Norton.

10

Part 1

Effecting Change

In the introduction we provided an account of the approaches that were used inthe first three phases of curriculum reform, as described by the McIntosh model,to increase the participation of women in mathematics. This part of the bookfocuses on transitional approaches (from Phase Three Women as a problem inmathematics to Phase Four Women as central to mathematics). In theirattempt to change the attitudes and behaviours of women towards mathematics wesee many features typical of Phase Three approaches. However, the approachesincluded here, to different degrees, are distinguished by the extent to which theyquestion monocuhural assumptions and/or arc influenced by some form offeminism. None of these approaches is firmly rooted in a Phase Three deficitphilosophy that identifies gender with women and sees women as victims inmathematics education with problems for which they need help. Rather, theseapproaches seek to redress the gender imbalance in mathematics, education bylocating its causes outside women. The chapters of this part are organizedaccording to their primary focus: female students, future teachers, practisingteachers and the public.

Two mathematics camps described in this section intervene exclusively withfemale students and have as a common starting point the goal of improving girls'chances to succeed in mathematics by teaching them skills of survival. 'Summer-Math', the summer learning community for secondary school girls described byCharlene and James Morrow, is designed to address the lack of challenge andsupport ficed by many female students in learning mathematics. Drawing onconstructivist learning principles and feminist scholarship on connected teaching,the programme provides a variety of experiences that connect girls to mathematicsrather than alienating them from it. Denisse Thompson's chapter describes the'Metro Achievement Program' which, like SummerMath, began as a summeroption and has since expanded into a year-round multi-faceted interventionprogramme designed to enable disadvantaged girls of average ability reach their fullpotential, both academically and personally. Working outside the school environ-ment, the programme augments the regular school programme by providing girlswith more challenging and relevant mathematics experiences and opportunities totake risks and assume leadership roles.

Olive Fullerton and Lynn Friedman both focus on working with futureteachers but their approaches derive from very differeqt Liunist perspectives andare situated in different phases of the McIntosh model. Fullerton's chapter detailsher work with a small group of prospective elementary teachers. In jointdiscussion, they reflected on their experiences in learning mathematics and the

11

Effiyting Change

anxiety they shared. These reflections revealed, among others, that lack ofexperience and facility in using mathematical language and technical terminologywere the main reasons these future teachers felt intimidated during mathematicslessons arid as a result lost confidence in their mathematical abilities. Fullerton

makes a series of recommendations, characteristic of feMinism of difference (see

Mura, Chapter 19), for changing the teaching pmcess to benefit women's learningstyles. This places her approach in Phase Four in contrast to that of LynnFriedman's study which is in the transitional phase described in the introduction.Friedman's work springs from a review of recent studies and accounts which

document that the proportions of women receiving graduate degrees in mathe-matics, and hence the proportion of women teaching in post-secondary institu-

tions, are still too small. As conditions which foster success for women in gradUate

programmes, Friedman cites providing sensitive counselling, ensuring a criticalmass of women students in graduate schools of mathematics, and having'sympathetic and concerned women on the faculty. In contrast with Fullerton'sapproach, Friedman is influenced by liberal feminism in that she believes that by

changing the conditions of women in graduate programmes (though notably not

the women), the problem of low participation will be solved.Three chapters place work with practising teachers in the foreground. Although

interventionist by design, none of these approaches sees women as a problem inmathematics education. Rather they question the whole teaching and learning

process. The first two papers describe the establishment of networks ofeducators andother adults who play a key role in the lives of female students. The Swedish networkdescribed by Barbro Grevholm uses the medium of conferences and newsletters toshare information such as current research and teaching ideas, and to support and

encourage members to work on changing the gender balance in mathematicseducation. The Quebec 'Women Do Math' network developed by Claudie Solar.

Louise Lafortune and Helene Kayler, has similar aims, but in addition promotesteach;ng strategies and activities based on principles offeminist pedagogy The focusof Corneha Niederdrenk-Felgner's chapter is diffffent. In working with in-serviceteachers she aims to increase their awareness ofgender issues in the use oftechnologyand the need to change the whole education system.

An entirely different approach is taken by Mary Harris in the final chapterin

this section of the book. Her chapter describes the development of an exhibition,entitled 'Common Threads', which features the mathematics involved in women'straditional work. In its questioning of the conventional undervaluing of women'slabour, usually labelled not-mathematicalind in her focus on uncovering the

'real' mathematics of the workplace, Harris's work is illustrative of Phase Four

reform.

12

Chapter 2

Connecting Women with Mathematics

Charlene Morrow and James Morrow

A Summer Journey

Imagine yourself, a 15-year-old in the process of completing your second year inhigh-school, and one of six young women in a class of sixteen advanced-levelmathematics students. One day, a brochure for something called 'SummerMath'arrives in the mail. Looking through it, first at the pictures, you find these images:

three young womon iii-asuring a slab of rock;two students discussing what appears on a computer screen;two students (one with 1 er shoes off) studying next to a blackboard;two students putting a ro )ot together;a basketball court with a harci-driving woman dribbling toward the hoop;women together on a mountain top;four young women in a canoe; andtwo students intently at work in a laboratory.

Jumping out at you from the brochure appear these words from a former student:1 learned to think "why" and "how" and not just to do as I've always been told.I only wish I had asked those questions of my teachers in high-school but I willwhen I enter Mount Holyoke'. Juxtapose these words with what you heard fromyour teacher in school today, 'Just do Also staring you in the face are the words,'an opportunity for young women'.

Well, that brochure came at the right timc, and you sent off the application,sweating through the writing of an essay to gain admission. 'Good practice forapplying to college', your parents iisai. nvet' the next several weeks, forms andinformation arrive from SummerMath preparing you for the summer away fromhome for a 'college-like' experience.

The young woman beginning the journey described above is just one of overWOO young women we have worked intensively with during the twelve-yearhistory of the SummerMaih programme at Mount Holyoke College. After a

summer of constructing solutions to mathemati c. al problems and learning toexplain their problem-solving processes, many of these young women have takencharge of their mathematics learning in a variety of effective ways. Below we


describe the structure and pedagogy of Summer Math and discuss thc deepconnections between our approach and new scholarship on women's education(Belenky et al., 1986).

We 'begin our discussion with a brief summary of the state of mathematicseducation for women in the US. It is important to note that the body of literaturefrom which the following information is drawn has focused largely on majoritywomen. Clewell and Anderson (1991) have found that 'studies of women overlookwomen of color and that studies of students of color de-emphasize genderdifferences' (p. 3). While there are many parallels between the experiences ofwomen of colour and white women, the nature of the similarities and differenceshas not been carefully studied. This issue is central to our concerns because theSummerMath programme serves students from many racial backgrounds.

Why 'An Opportunity for Young Women'?

Two decades of research on gender issues in mathematics education in the US have

made it clear that there are still firm barriers preventing women's equalparticipation in mathematical studies and careers (Fennema and Leder, 1990;American Association of University Women, 1992; Sadker and Sadker, 1994). A

recent publication, Winning Women Into Mathematics (Kenschafi, 1991), puts fortha detailed and convincing case that, as a culture, we still do a great deal todiscourage women from participating in mathematics. Those individuals who have

taken a careful look at the ways that girls experience our educational system tell

us that girls are still receiving strong messages of exclusion, subordination, andobjectification. Even in elementary school, where girls generally receive bettergrades than boys, we begin to see the effects of these barriers. Girls begin toperform more poorly on higher-level cognitive tasks in mathematics. Moreover,this difference seems to pusist and grow through high-school (Hyde, Fennemaand Lamon, 1990).

Across all grade levels, the attention afforded to girls in the classroom is far less

than that given to boys (Leder, 1990; Sadker and Sadker, 1994). One particularaspect of this differential attention is that girls are not pushed to struggle forunderstanding. Usually, if a female cannot respond immediately to a question orproblem, she is not offered guidance or encouragement by the teacher to persistin her thinking.

Women are confronted by many barriers that men do not have to face. Forexample, a disturbing number of students agree with the statement, 'Mathematicsis more for boys than for girls'. Some studies found that such an attitude ispervasive among young males and that it holds, though to a lesser extent, amongyoung females (Eccles and Yee, 1988; Morrow and Morrow, 1991). In addition,this research has also demonstrated that many high-school counsellors still

discourage women from taking advanced mathematics courses and from consider-ing technological careers, that teachers interact more with their male students thanwith their female ones that boys, far more than girls, utilize computer resources

/4

Connecting Women with Mathematics

during unstructured time, and that parents still consider mathematics, science, andtechnology to be more suitable subjects for males than for females.

In advanced high-school mathematics classes, girls begin to be in the minorit7(Czujko and Be.nstein, 1989; Fennema, 1987). In college, despite the fact thatwomen represent about half of the mathematics majors, they still find themselvesin the minority in most of their mathematics classes due to the presence of a largenumber of men from male-dominated fields such as physics and engineering. Ingraduate school, the number of women declines drastically, to the point that menoutnumber women four to one at the PhD level in mathematics Millard, 1991).

Even for women who have chosen to dedicate themselves to mathematics,many barriers remain in place, especially at the most prestigious researchuniversities where only 5 per cent of the entire mathematics faculty are female(Jackson, 1991). Indeed, successful educational environments for women are moreoften found in the liberal arts colleges than in the research universities. Liberal artscolleges, particularly women's colleges, contribute disproportionately to femalePhDs in mathematics and engineering (Sharpe, 1992).

Career statistics for women indicate a vast disparity between the se \es inparticipation rates in quantitatively based fields (National Research Council,1991). The more mathematics required in a particular career, the higher the payand lower the rate of female involvement. The career picture for women in scienceand engineering fields is especially bleak. As of 1984, women, who represent 43per cent of all professional workers, held only about 12.8 per cent of the jobs inscience and engineering. Approximately 1 per cent of engineers, 2 per cent ofphysicists, and 5 per cent of cheinists are women (National Research Council,1991). The situation for women of colour is even worse (Clewell and Anderson,1991).

Until very recently, female role models have consistently been written out ofthe history of mathematics (Perl, 1978; Perl and Manning, 1985). The one female,Emmy Noether, who is included in the 'Men of Mathematics' poster (producedby IBM and covering over 2000 years of history in mathematics) is devilishly hardto find. The culture of mathematics remains distant, cold, and undesirable for toomany women.

The research literature is quite consistent regarding the factors associated withwomen's success in mathematics classrooms. Eccles (1987) describes 'girl-friendly'classrooms as having 'low levels of competition, high levels of cooperative learningor individualistic learning structure, high levels of teacher communication ... ofthe intrinsic value of maths and the link between maths and various interestingoccupations' (Eccles, 1987, p. 158). Pedersen (1988) adds that classrooms that aresuccessful for females are 'not remedial ... and employ a problem solving formatin which students work on challenging problems' (Pedersen, 1988, pp. 209-10).Unfortunately, girls have rarely had the opportunity to reap the benefits of thesefindings. We, along with many others, conclude that a multitude of social factorsdiscourage women from making mathematics a significant part of their lives.

15

4:- 0

Charlene Morrow andJames Morrow

Summer Math: Reform within a Feminist Framework

Let us now return to the summer journey with which we began our discussion, tothe programme called SunmierMath, held each summer on the campus ofMount Holyoke College. SummerMath is an intensive, six-week residentialmathematics learning community for 100 high-school women. It is designed toaddress the ways in which women are underserved in mathem4tically based fields byproviding new perspectives and new experiences of mathematics, computing, andscience. The student body is very diverse, racially, geographically, and academicallyIn 1994, our student body Was 37 per cent African or African American, 17 per centLatina, 13 per cent Asian, 2 per cent Native American, and 30 per cent EuropeanAmerican. They came from all across the mainland US, Hawai'i, Puerto Rico, Japan,Paraguay, Brazil, and Pakistan with ages ranging from 13 to 18 years. Some of ourstudents have achieved good grades in mathematics, while others have not, butalmost all feel the downward spiralling ofself-esteem and confidence in mathematicsthat is so characteristic of high-school women (Eccles, 1987; American Associationof University Women, 1992). We emphasize greater conceptual understanding,affirmation of young women as capable members of a learning community, and theimportance of constructing one's own understanding of complex ideas.

The instructional methods, based on constructivist (von Glasersfeld, 1983) andconnected models oflearning (Belenky et al., 1986), are designed to change the blindmemorizing and rule-following behaviours that limit so many students' under-standing of the underlying concepts of mathematics. SununerMath helps studentsreplace such unproductive methods with more flexible problem-solving approaches.The atmosphere of the programme is one of challenge and support. That is, thechallenge of rigorous study and difficult problems, coupled with the support of acommunity of teachers, residential staff, and peers. Our goal is to help studentsacquire ;?- :met confidence in their ability to excel, greater persistence in problem-solving environments, and a greater conceptual understanding of mathematics.

Stair Preparation

For one week before the students arrive, our staff of thirty-five meets to prepare.Our activities fall into three categories: doing mathematical and computeractivities; training teachers, teaching assistants, 1:nd residence assistants; andconsciousness-raising around issues such as racial and gender equity It also includespractice and reflection on mathematical activities in order to build understandingand commitment to constructivist principles of active learning. Our focus is tolook within ourselves for potential points of connection that can fostei the growthand development Of our students.

Cl(bes

Students at SummerMath participate each day in cla,scs characterized by studentactivity, questioning. discussion, and discovery In all sessions, pupils begin by

/ 6

Connecting 144micit with Mathematics

either solving problems, or posing their own problems. They experience a newvision of mathematics through two classes, 'Fundamental Mathematical Concepts'and 'Computer Programming' (LOGO), and three workshops.

'Fundamental Mathematical Concepts' is the heart of SummerMath. Studentswork and discuss ideas in pairs or small groups, with at least one instructor and oneundergraduate assistant circulating about the classroom posing questions andfacilitating discussion. The specially designed problem-based curriculum focuseson concepts essential to understanding advanced mathematics and science such asratio, linear relationships, functions, patterns, and logic. Students are encouragedto assume an authoritative role as they construct their own understanding ofmathematical concepts. Visual and verbal methods of illustration, as well asnumerical, algebraic, and graphical representation, are stressed as means to deeperunderstanding. The frequent discussion among students, as well as bctweenstudents and their instructors, provides a supportive environment that encouragesreflection on thinking and the stimulus of collaborative effort.

In LOGO, students work in pairs at a computer and solve problems in thecontext of design projects. They pursue mathematical ideas ranging fromelementary geometry to recursive functions. They learn to plan, organize, andrevise their ideas by working on tasks such as transformational geometry, tangrampuziles. patchwork quilt designs, and group murals. In the last week ofSommerMath, students demonstrate the programming skills they have developedby collaboratively designing, developing, and completing a major computer project.

Finally, each student takes three two-week workshops. choosing froma varietyof options s.,:ch as confidence building, brain imaging, the art ofmaking anatomicalcomparisons, architecture, gcnetics, economics, psychology, and the physics ofmotion. These workshops provide experiences that place students in the role ofscientists, thinking up the questions to be asked, performing experiments, collect-ing and anal,sing data, and comparing their observations with their hypotheses.

The Residential Community

The residential component of the programme is crucial in the creation of a strong,close-knit community at SummerMath. The residential staff is headed by acoordinator and an assistant, and includes several undergraduate residentialassistants (RAs) who are also teaching assistants. The RAs give SummerMathstudents support and friendship as they flee the challenges of learning andexplaining their ideas in depth. In consultation with Summe-Math participants, a

series of workshops and social activities is developed, including issues of sexuality,nutrition and other health-related topics, dance and music, and t-shirt decorationand beadwork. Students are offered many additional activities intended to linktheir mathematics experience to the outside world, to give them a realistic viewof career optionsind to broaden their horizons. Once a week, speakersrepresenting science, business, and various professions are invited to present anevening lecture, panel discussion, or laboratory demonstration and answerquestions about their work.

r, 17

Charkne Morrow and James Morrow

Assessment

Assessment is embedded in the learning process through ongoing discussions withSummer Math staff. Students evaluate their own progress half way through theprogramme, and discuss their self-evaluation with their instructors. At the end ofSummer Math, teachers comment on the progress of the pupils in their classes andmake suggestions for further work.

Students write an extensive final evaluation of their summer experience,including feedback on all aspects of the progranune. This evaluation is both anopportunity for each student to reflect on her own experiences, as well as a chancefor us to learn how to improve the programme in the future. During the academicyear following the programme, students complete a questionnaire investigatingpossible changes in attitude and new ways of approaching learning, as well asproviding information about the courses they are taking, their grades, andstandardized test scores.

Avenues of Connection for Women

1Vomen's l/Vays of Knowim (Belenky et al., 1986) gives a compelling description ofthe manner in which teaching can alienate women from most academic areas.Meaningful educational experiences described by women are those which permitthem to weave together multiple aspects of their lives. The authors name this kindof educational experience 'connected learning', in that students are encouraged tobuild on their entire knowledge base rather than leaving all personal experiencesat the classroom door. A teacher who facilitates this kind of educauon is engagingin connected teaching.

The following sections describe the journey from disconnection to connec-tion in mathematics that we have observed many times in our students. Eachsection represents a critical issue for females and each is an integral component ofour approach. These overarching issues are drawn largely from Mmen's Wiys ofKnowing. Within each segment, describe the type of approach used in ourprogramme to implement cur educational goals.

Cogfirmation of Self in the Learn* Community

Perhaps one of the most critical issues in education is that many women describethemselves as remaining on the fringes of the intellectual community. Inmathematics, because women often feel completely on the outside, it is

particularly important to structure the classroom as a setting with a place foreveryone, where it is clear from the beginning that what students already knowwill be useful, that they will be treated as capable learners, and that there are noprerequisites for belonging.

At SummerMath. one way in which the self is confirmed is by having studentswrite about and share problem-solving experiences and feelings (Buerk, 1985;

18

Comm-ring Women with Mathematics

Morrow and Schifter, 1988). In this way, students come to see that there are manyacceptable paths to knowing in mathematics. The teacher's role as facilitator andguide, rather than lecturer and expert, is critical to this process. The teacher mustbecome skilled in active listening and asking questions that will allow the studentto become more aware of her own thinking, as well as to decide which of her ideasto pursue further. This kind of teacher role has been likened to that of the Socraticteacher. However, a crucial difference for us is that the Socratic teacher maintainscontrol of the dialogue mainly by asking leading questions and defining the termsof the conversation. At SummerMath, the student is in charge of the interactionand thus, the direction of talk is primarily defined by the pupil. The teacher, ofcourse, inevitably provides a 'safety net', whereby misconceptions can be recog-nized and addressed. But, when the student is in control, rather than the teacher, it ismuch clearer to both parties when conceptual understanding has been attained.

Learniv in the Believing Mode qf Communication and Qh:'stioning

Many of the women interviewed by Belenky and her colleagues expressed adistaste for being in an argumentative atmosphere and, th,:refore, would oftenpatiently await its end rather than participate. As a result., inquiry that is based onbelief, rather than doubt, is essential to a classroom environment that serveswomen well. As our students share problem-solving experiences, it becomes thecole of the teacher and other students to question why and to ask for details: Whydoes such a step make sense? What part of the problem allows for such a statement?What more can be said about the process used? Such questions are asked in thebelief that the student has a valuable understanding that can be expanded, clarified,and modified.

Students are asked to provide detailed explanations for all solutions toproblems through an inquiry process based on belief in the student rather thandisbelief of what the student says. As each individual becomes used to explainingher thoughts and processes, she becomes more and more of a mathematician anda creator of mathematical knowledge. When a student can demonstrate to herselfthat she knows why a solution works, she becomes more confident and lessteacher-dependent.

Ming on Challenges with Support

In order to grow intellectually, a student must take reasonable risks and be able tomake mistakes. We, as a culture, have a difficult tnile allowing risk-takingbehaviours in women. We 'rescue' females at the smallest sign of discomfort andhelp them into helplessness by providing the kind of support that takes controlaway from the individual. On the other hand, we give all students the message thatyou 'have to be tough', 'gut it out', or 'just do it' in order to get ahead. Thesemessages convey challenge with no support and a sense of isolation, creating anatmosphere that many women avoid.

At SummerMath, our students work in pairs in all parts of the programme,

1/4 .

19


including computer tasks. In the beginning, each student in the pair is given a

specific role, either the problem-solver, or the active-listener/questioner in thebelieving mode described above. Students switch roles as they move from oneproblem to the next. In this way, they become better both at persisting in problem-solving, and in asking themselves questions that will help them to move along.

Pupils also become better at asking specific and focused questions of teachers whenhelp is needed. In order to become confident learners, students must struggle forunderstanding. However, they can struggle together and look to each other and

the teacher for support and encouragement.

The Development qf Voice

To help students gain a sense of their own voice in mathematics, we first help them

move away from a strictly answer-oriented approach to solving problems. We havefound that it is impossible to achieve this goal in an environment where answers aresupplied. Therefore, we do not tell students whether they are right or wrong. Nor arcstudents told how to solve a problem. When neither answers nor instructions aresupplied, students are forced to look to themselves and to each other in order todevise a strategy for solving the problem. In explaining her solutions, the studentmust learn to give voice to her discoveries. We emphasize visual representations ofsolutions as an effective mediator between a student's ideas nd her expressionoftvem. Participants report that being able to 'see what lam doing' is a very powerfulexperience. The student can then give voice to what she sees.

Becomhig a GmstriKtor qf Knowledge

As students work through mathematical challenges in a supportive learningcommunity and develop a sense of voice and authority in the classroom, theybegin to define their own questions. They become excited about the possibilitiesof posing their own problems and inventing new knowledge. They no longerremain outside, but become part of the inner circle of knowers, with their ownpower base. As this awakening begins, we try to provide insights into mathematicalconnections with other areas. Workshops in mathematically related fields such asarchitecture, chemistry, and medicine help students learn about the manner inwhich mathematics is applied in a variety of settings.

20

The Journey Home

A few 1,;ears after attending SummerMath, 1 enrolled in a college calculuscourse. In the timl exam. I looked at the problems and quickly realizedthat I thought I could do all of the problems, but I would not be able tofinish in the time allotted. I set to work, recalling the ways in which Ilearned to work confidently at SununerMath. When the time came tohand in papers, 1, of course, was not done. I refused to give the professor

Conniving amen with Mathematics

my paper and continued to work. I worked and worked until I hadfinished all of the problems, and felt confident that I had done themcorrectly. I handed my paper to my professor and said, 'I dare you tome'. I received an A! (Danielle)

We have heard many other stories of young women taking charge of their learning,from simply asking the kinds of questions that enable them to move ahead, tohaving the courage to take honours and advanced level courses. The data we havecollected on attitude changes experienced by our students (Morrow, 1991)indicate that our pupils leave Summer Math feeling more able to persist in doingmathematics problems, more aware of the usefulness of mathematics, anddramatically more confident. Changes in student attitudes, beliefs, and behaviourare most effectively illustrated by students' own comments on their abilities tolearn mathematics.

On beliefs about oneself:

I think I have learned new techniques for solving problems, and havebecome more secure and confident. Now I know why certain theorems,etc. work.I learned you don't have to be a Mister Wizard to solve a confusinglyworded problem.It makes me now, believe it or not, like problems that are hard. It also gaveme patience and several ways to solve a problem. My thought processes willnever be the same.I learned more about myself. I learned it's alright to speak up. It gave mecourage and made me more relaxed in all sorts of situations.

On working with a partner:

You get to see how other people think and how they're taught.You don't feel alone.Having to justify all of your answers.

On things that will be difThrent in mathematics class:

I'll feel I can depend on myself.The fact that I know there is more than one way to work a problem, if Idon't understand the given way.I won't give up as easily when I get to a difficult problem; I will try differentways to solve problems instead of one way; I will strive more for goodgrades because I see how important mathematics is.

On experiences in fimdamental mathematical concepts:

I feel I'm capable of solving almost any math problem and if I stick in thereI'll solve the problem, happily.I learned a lot about math, people, and the real world, good and bad.Self-satisfaction; better understanding; less frustrated when I don't under-

21


stand everything the first time.I learned a great deal about how I think about mathematics and solveproblems, about the logic behind math procedures and how to rely onmyself to solve problems. I also learned how to listen more carefully to myown ideas and how to find what my weaknesses are.

These comments illustrate that students take home with them new ways toapproach mathematics that emphasize reasoning, comfort with the subject,motivation to study it further, a better understanding of certain fundamental

concepts, and an overall increased self-confidence. They become independent, butnot isolated, problem-solvers, retaining a sense of authority within themselves, butable to effectively communicate with others about mathematics.

An additional benefit participants gain is a much clearer sense of what a college

environment is like and what kind of career choices may lie ahead. Our research

on programme outcomes and resulting attitude changes (Morrow, 1991) indicatesthat SummerMath has proved extremely effective in furthering the mathematicaland career aspirations of students from widely varying backgrounds, not just themathematically precocious. Furthermore, analysis of data by racial category has

shown that SummerMath has been effective for students of colour as well as white

students, both in goals achieved during the summer and in positive attitude

changes.In such a strong, female-student community, intellectual and social compo-

nents of friendships are strongly linked. Women don't have to live up to theexpectations they believe males have of them. In SummerMath classrooms, there

is an atmosphere of trust, cooperation, and support where women's voices arealways heard, respected, and responded to. Teachers maintain high standards andpush students to struggle for understanding in the belief that all pupils can use theirpersonal experience effifctively as well as justify their own ideas. In this atmosphereof belief, pupils become aware of, and trust, their own intellectual authority, andbegin to construct their own knowledge and persist in problem-solving tasks. As

an individual in the 1992 programme said,

I learned so much about myself and how I interact with thoughts on aregular basis. I know I can be happy with myself and as long as I have thatconfidence in myself, I can take on challenges and not worry so muchabout failure. I will know that as long as I am thinking and working, I amsucceeding.

The Case for a Single-Sex Mathematics Programme

Our belief in the virtue of a single-sex mathematics programme for girls isgrounded in the long tradition of women's colleges, of which our institution,Mount Holyoke College, is a prime example. There is solid evidence (Riordan,1990; Tidball, 1980) that women's educational institutions provide an atmospherein which young women more successfully test their skills, find their voices, and go

22

Connectiv Wwnett with Mathematics

on to participate in a wide variety of arenas, including many that have beentraditionally male-dominated.

It is interesting to note, however, that the mathematics education provided inthe Summer Math learning community is not only in line with the structure of`girl-friendly classrooms' described in the literature (Eccles, 1987), it is alsoconsistent with that now recommended by the National Council of Teachers ofMathematic:, (NCTM, 1989) for all students in the US. The advent of newstandards in secondary mathematics teaching, and the close attention being paidto the teaching of calculus at the college level (Steen, 1988), have brought abouta great deal of positive change in mathematics education. There is now anemphasis on understanding rather than just memorizing, on communicatingmathematics, and on doing mathematics in context. The publication of Pro-fessional Standards for Teaching Mathematics, by the NCTM in 1991, introducedthe expectation that all secondary mathematics teachers, as well as teachereducators, will use much of the pedagogy we have been using at SummerMath.Why then, are we continuing to promote an all-female programme?

The fiict is that pedagogical innovation in a coeducational environment doesnot automatically guarantee gender equity for the reasons discussed at thebeginning of this chapter. This point is clearly illustrated by the following story ofa middle-school teacher engaged in implementing sonic very positive changes inher classroom that arc in line with the new NCTM teaching standards:

Yet, even as this new atmosphere established itself in the classroom, Lisabecame aware that some negative social norms continued to operate.Most dramatically, in this class of eight girls and fifteen boys, Lisa realized,several months into the year, that the girls in the class had no voice.Gender patterns that had been established in previous years of mathe-matics instruction, patterns reinforced by societal attitudes, continued tointrude ... Lisa had established a classroom dynamic of active andsometimes excited exchange of mathematical ideas, but the boys hadcome to dominate discussion. The girls were reticent about participatingand thus were marginalized.

Lisa saw a way to address the problem when, during a discussion with.1 mother who was concerned that her daughter had already given up onmathematics, Lisa suggested that her daughter and a friend stay after classon Wednesdays to do mathematics. But then, on second thought, shedecided to encourage all eight girls to conie, and they formed the 'girls'maths club'. The formation of the girls' maths club significantly alteredthe classroom dynamic. Lisa says, 'What the girls mastered ... changed ...their voice in the class. The voice of the classroom had been extremelymasculine and what they did was to change the quality of that voice'.(Schifter and Fosnot, 1993, pp. 135-6)

Fortunately, this teacher could recognize and respond efkctively to the traditionalgender roles that remained in place despite her new pedagogical approaches. Given

2.3

Charlene Morrow andJarnes Alorrow

the strength with which gender roles resist change, we have little reason to believe

that most educators will be able to see and intervene in this fiishion without a greatdeal of encouragement and support. Teachers are rarely trained in explicitstrategies for recognizing gender inequities or in the detailed approaches necessaryto overcome them.. It is one thing, for instance, to have an expectation forcollaborative classrooms, but it is quite another to ensure that mixed-sexcooperative groups do not simply become another place where males do thetalking and doing, while females do the listening and recording.

We do not wish to make the argument here that a single-sex environment is

always best, only that it can and does serve young women well in some instances,particularly when gender messages are as strong as they have been regardingparticipation in mathematics and sciences. Neither do we wish to suggest that asingle-sex environment is automatically an atmosphere in which young womencan grow to their full potential. Historically, many women's institutions have beenbastions for the conservation of women's traditional roles, functioning only tokeep them in their place. To counteract unconscious conformity with societalexpectations, there must be a purposeful effort to challenge gender stereotyping.

Strategies for Change

To make 'mathematics for all' a reahty for women. we offer the tbllowing strategiesbased on our experiences with students and staff at SummerMath. While webelieve that these approaches can often be implemented more effectively in asingle-sex setting, we suggest that they are important considerations in anyeducational environment attempting to serve females well.

24

Look at the experiences of female students on a regular basis. Being 'genderblind' is more likely to result in the continuation of inequitable patternsthan in equal treatnwnt.Look for the valuable aspects of female approaches to learning mathe-matics. It is too easy to fall into the trap of using successful male strategiesas the norm for 411.Develop a supportive professional climate for examining and challengingexisting gender roles. Changing the classroom climate must be a collabora-tive effbrt lw teachers and administrators. Educators in senior positionsmust be willing to assume a leadership role on this issue.Believe that female students thrive on intellectual challenge and that theydeserve to be supported in their efforts.Become familiar with new scholarship on women, such as that done byBelenky et al. Cross-disciplinary thinking is rarely used to envision a moregender-inclusive educational environment.Challenge the personal beliefs of students, tea( hersind administratorsabout the nature and usefulness of mathematics. Educating people aboutwhat mathematics is and does can create many new pathways for learning.

Connecting Mmen with Mathematics

Make sure that structures for small group work give students explicitstrategies for sharing time and tasks equitably. It is likely that inequitablegender-hased behaviour will continue and be reinforced in unstructuredsmall group work.

Conclusion

It is a deeply ingrained idea, even within the broader feminist community, thatmathematics is an inherently alienating experience for females. Many youngwomen view learning mathematics solely as a stepping-stone to another place, aplace where mathematics will seldom play a major role. We have too often heardthe comment from young women that, 'I can do math, and it's even funsometimes, but I want to choose a career that will allow me to do something usefulwith my life. I want to work with/for people.' Educators, parents, and mentorsoften impart the same view to young people, passing along the notion that, forinstance, literature and social science are richly connected to human concerns andcreativity, while mathematics is removed and mechanical. We believe that theSummer Math programme offers a model for helping young women see that theycan do mathematics confidently and that through mathematics they can becomeconnected to people rather than disconneaed from others.

References

AMI RI( AN A`,\OCI AI I( r`: UNIVI KM I y W )MI N (1992) HOW Schools Shorn-hank' Gir/s.Washington, DC, American Association of University Women Educational Founda-tion.

131 I I-NKY, M., Ci it IV, B., (oi I my ma-R. N. and TARut I , (1986) II ,ttlett's Hays ofKnowing, New York. Basic Books.

Bit i ARI), L. (1991) 'The past, present, and future of academic women in the mathematicalsciemes', Notkes tlw Anwrkan lathetnatical Society, 38, pp. 707-14.

By1.11K. D. (1985) 'The voices of women making meming in mathematics'. Journal ofEducation, 167. pp. 59-70.

Ci I WI I , B.C. and ANDi R.soN, B. (1991) Iliimen of Color in Mathematics, Science andEngineering: A Reriew the Literature, Wishington, DC, Center For Women PolicyStudies.

, R. and Ri P.Ns II IN, 1). (1989) 11 Ito TlheS Science? A Report on Student Course-linkin HO School Science and Mathematics, New York, American Institute of Physics.

E ii s, J. (1987) 'Gender roles and women's achievement-related decisions', Psydwlogy ofItnnen Quarterly, 11, pp. 135-72.

E«i t s, J. and Yi s , I). (1988) 'Parent perceptions and attributions for children's mathsachievement'. Sex Roles: A Journal of Research, 19, pp. 317-33.

Fi NNI MA, E. (1987) 'Sex related differences in education: Myths, realities. and inter-ventions'. in RI( I IMO /50N-KnI III I It, V. (Ed) The Educator's Handbook!: A ResearchPerspecth.e, Longman Press, pp. 329-47.

25

Li )


FENNPAA, E. and LEDLIk, G. (Eds) (1990) Mathematics and Gender. New York, TeachersCollege Press.

HyDE, J., FENNLMA, E. and LAMON, S. (1990) 'Gender differences in mathematicsperfornunce: A meta-analysis', Psychological Bulletin. 107, pp. 139-55.

JACKSON, A. (1991) 'Top producers of women mathematics doctorates'. Notices of theAmerican Mathematical Society, 38, pp. 715-20.

KENsctiAFT, P. (Ed) (1991) Winning Wonwn into Mathematics, Wishington, DC, Mathemat-ical Association of America.

LE0m. G. (1990) 'Gender and classroom practice'. in Bulk rots:. L. (Ed) Gender andMathematics: An International Perspective, London, Cassell Educational Limited,pp. 9-19.

LINN, M.C. and Hy J.S. (1989) 'Gender, nuthenuticsmd science'. EducationalResearcher, 18, pp. 17-27.

MoRkow. C. (1991. April) 'A description of Summer Math, a process-oriented mathe-matics learning comn1unity, and resulting student attitudes toward mathenutics'. Paperpresented to the National Council of Teachers of Mathematics, New Orleans, LA.

MoRkow, C. and MORROW, J. (1991, October) The SummerMath Program fin Minority Girlsand Twthers, Annual Report, National Aeronautics and Space Administration Project.

MoRkow; C. and Scull mt, D. (1988) 'Promoting good nuthematical thinking throughwriting in the classroom'. Proceedings of the 'Thnth Annual Psychology of MathematicsEducation-North An:erica, DeKalb, IL.

NA I IONAI COLIN( 'II OI TI Of MAI I II'MA I I( s (1989) Curriculum an). .1-1'valuationStandards for School athematics. Reston, VA, National Council s, Teachers ofMathematics.

NArioNAI COUNCII 01 TI.A( MRS ()I MA I I IIMA I IC', (1991) Prokssional Standards .for'Thaching Mathematics. Reston, VA, National Council of Teachers of Mathematics.

NA FR /NAI RP,I.ARCI I COUNCII (1991) It 'omen in Science and Engineering: Increasing theirNumbers in the I990s, Washington, DC. National Academy Press.

Pt DI.Rst-N. K. (1988) 'Combating the stereotype of women in mathematics and women'sstereotype of mathematics', Prowedings of The Matlwmaticians and Edumtion RefinmNetwork, National Science Foundation !finks/lop. Chicago. IL.

T. (1978) Math Equals. Menlo Park, Addison-Wesley.T. and MANNING, J. (1985) I timten, Numbers, and Dreams, Santa Rosa, CA, National

Women's History Project.RIORA mN, C. (199(1) Girls and Boys in School: Tigether or Sepahne?, New York, Teachers

College Press.SAi)o Is, M. and SADKFR, D. (1994) Failing at Fairness: How America's Schools Clwat Girls.

New York, Charles Scribner's Sons.Sct m Illk , D. and FosNm. C.T. (1993) Reconstructing Mathematics Education: Stories of

ii.achers Aleeting the Chalknge Refinm, New York, Teachers College Press.N.R. (1992). 'Pipeline from small colleges'. Notices of the American Mathematical

Society 39, p. 3.SIIIN, L.A. (Ed) (1988) Calculus fin a New Century: Pump, not a Fihm MAA Notes No.8,

Washington, DC, Mathematical Association of America.Tii mAi 1, E. (198(1) 'Women's colleges and women achievers revisited', Signs: Journal of

fi'omen in Culture and Society 5, pp. 104-10.voN Ci Am toil] it, E. (1983) leltning as a constructivist activity', Proceedinp of the Hilt

Annual 3 1ce6tw of Plchokgy of :I fathematics Ethwation-North America, Montr6al,Canada.

26

Chapter 3

The METRO Achievement Program:Helping Inner-City Girls Excel

Denisse R. Thompson

Introduction

In the List decade, numerous reports have signalled the need to increase theparticipation in mathematics of students in North America (MathematicalSciences Education Board, 1990; National Council of Teachers of Mathematics,1989.1991; National Research Council, 1989). These reports have indicated that.from high-school to graduate school, roughly half of all students at a particularlevel do not proceed to the next. For example. while approximately 50 per centof all high-school students take more than two years of college-preparatorymathematics, only about 25 per cent take more than three years of mathematicsat this level. For female students the problem is especially acute, since women inthe US have traditionally taken fewer mathematics and science courses than males(Campbell, 1)86; Meyer. 1989). However, by the year 2000 it is predicted thatone-third of the total US workforceis well as 85 per cent of new eni7::.as to theworkforce, will be women. minorities. and immigrants. If the workforce is to bea viable one, these groups need more and better mathematical training (Johnstonand Packer, 1987).

Clearly, there is a need for broadly-based programmes that encourage allstudents to develop their abilities to their fullest potential. However, there arepopulations whose particular needs may best be served by programmes whichfocus specifically on their concerns. First, programmes serving female studentsshould be developed which .encourage them to keep their future options open bycontinuing with academic courses and this should begin early, tbr example,between 11 and 13 years of age. Nelson (1986) indicates that this is the period'when girls are most vulnerable to peer values about what it means to becompetent, successtUl. and female. Classroom and out-of-school experienceswhich consistently conmiunicate stereotyped or limited expectations or are sparsein stimulating examples and encouragement will undercut girls' motivation tolearn and will undercut the development of their full potential as productive andself-actualizing adults' (pp. 3-4). It is also at this age that girls' confidence in their

27

Denisse R. Thompson

own abilities and their expectations of success begin to decrease, factors whichhave been found (Campbell, 1986) to precede the decline in mathematicsachievement by girls. These are the years when, according to Nicholson (1986),patterns of disengaging with mathematics and science become established. Thus,we need to design programmes to nurture female students. In single-sexenvironments female students are forced to assume leadership roles and developcooperative networks. They cannot `bow out' of difficult problems leaving themto the boys and so have to take the lead and solve problems for themselves.

The second population we need to target specifically are students of averageability. In the US, special education programmes are provided for students whohave demonstrated strong academic promise and for those who are struggling andneed remediation; few programmes exist to provide additional support for thelarge group of students in the middle. By encouraging those of average ability, itmay be possible to foster links to strong academic programmes and keep suchstudents in mathematics and science.

Third, in the US there is a need for programmes which serve students frominner-city neighbourhoods. These environments are home to a variety of racialand ethnic minority groups and much of the population is of a low socio-economic status. The schools often lack modern facilities and current materials(Matthews, 1984; McBay, 1990) and have high attrition rates (Arias, 1986).Therefore, support programmes are needed to enhance the educational experienceof inner-city students and encourage them to remain in the system and to takeadvantage of academic opportunities.

The 'METRO Achievement Program" was designed to address all three ofthese needs. Designed as an outside-school activity, the programme serves femalestudents of average ability from the west side of Chicago. The programme aims toencourage academic achievement, especially in mathematics and science, to aidstudents' personal development, and to instill a spirit of community service. Thischapter describes the programme and presents data on participants' perceptions ofits benefits as well as their views on the programme being restricted to females.

Description of the Programme

The METRO Achievement Program began in 1985 as a summer progranmie forgirls of average ability who were entering the seventh (age 12 years) or eighth (age13 years) grades in the following fall. The programme was modelled on one forboys of the same age which had been operating in the area since 1962. Initially atutoring and sports facility it was expanded in 1972 to emphasize academicachievement and character development. When it was realized that girls neededsimilar opportunities, the founders made the decision not to make the existingprogramme coeducational since they felt that more could be accomplished withadolescents in a single-sex environment.

Although still focusing primarily on the middle-school grades 7 and 8, theMETRO Achievement Program has expanded since 1985 to include activities

28

The METRO Achievement Program

throughout the academic year for girls in 6rades 4-12 (ages 9-17 years). At thepresent time, the programme consist: of three segi-n--nts. For girls between 9 and11 years of age, there is an individual tutoring programme throughout the schoolyear. Approximately 60 per cent of the tutors are professional women, 20 per centare female college students, Ind the remaining 20 per cent are girls from the high-school METRO Program who devote one day per week to this activity. Thisprovides the young girls not only with the tutoring they need but also with femalerole models. In the summer, girls in this age group may participate in a one-weekmini-version of the programme offered to the older girls.

For 12- and 13-year-old girls, there is a five-week summer programme thatconcentrates on developing the whole person. Girls participate in a variety ofacademic courses, for example, mathematics with an emphasis on applications andtechnology, science centred on laboratory work, and communication skillsincluding writing, grammarand public-speaking. As well, girls are offered sessionson fine arts, team sports, personal counselling, and character development with afocus on interpersonal relationships. During the regular school year, participantscomplete a programme that is similar to that offered in the summer. Designed toupgrade skills, this programme aims to encourage students to enrol in an academictrack rather than a general track with no college-preparatory courses.

At the high-school level, ages 14-17 years, the emphasis is on keeping girlsin school and promoting enrolment in a college or university. The programme forthe youngest girls concentrates on public-speaking as well as on the study ofalgebra emphasizing the use of graphing technology, graphics calculators andcomputer-graphing software. The curriculum for the following var focuses onwriting skills using word-processing applicationsind on the development of visualthinking skills using computer-generated geometric drawing tools. The 16-year-olds begin preparation for college-entrance exams, and in the final year theconcentration is on the study of world culture. These curricular emphases closelyfollow the areas of study typically found in US high schools at this level.

The METRO Program recruits participants by making end-of-the-year visitsto schools within its service area. After completing a survey indicating their interestin the programme, girls are chosen on the basis of follow-up interviews withapplicants and their parents. During the selection interview, METRO tries toascertain whether the girls' attitudes are in tune with the programme's goals. Staffalso review their academic and attendance records, looking particularly atconsistency of grades, comments on behaviour, lateness and absenteeism Standar-dized test scores are also examined, with girls generally needing a score at least inthe fortieth percentile. Essentially, the staff try to identitY those good students whojust need a chance. For a more detailed description of the selection process seeThompson and jakucyn (1993).

During the summer, the programme serves about sixty 9-11-year-olds 'andeighty 12-13-year-okls. During the school year, there are approximately ninetygirls ag( d 9-11 years, eighty aged 12-13 years, and eighty in the oldest age group(ages 14-17 years). The overall omposition of the METRO Program intake is acritical feature. The diversity of the student population (6)) per cent Hispanic, 30

29

.

Denisse R.. Thompson

per cent African-American, and 10 per cent Caucasian or Asian) gives girls theopportunity to interact with members of other racial or ethnic groups. For manystudents, this is the first time they have had the chance to make a friend withsomeone from another race.

Mathematics Component

The programme for the 12-14-year-old students uses materials developed by theUniversity of Chicago School Mathematics Project (Usiskin, 1988, 1991). Thesematerials were developed to provide students of average ability with a broaderrange of topics than is traditionally found in American textbooks. Topics added

include statistic, probability, discrete mathematics, technology and applicatic ,s.Students begin with applied arithmetic and then move on to topics in algebra (suchas patterning, variables, open sentences), pre-geometry (angles, polygons, area,perimeter), and more advanced problems in applied arithmetic.

There are several features that make these materials appropriate for use withthe programme. First, students are expected to use scientific calculators, and theseare made available to all participants in the programme. This ensures that lack ofcomputational facility does not prevent the girls from studying mathematical topicsthey nuy not yet have met in their regular school curriculum. Further, thepresence of technology allows for a focus on real-world problems with realisticnumbets.

Second, 'real world' applications are integrated throughout the curriculum sothat participants realize the importance of mathematics and its applicability to awide range of situations. Applications are used to link branches of mathematics,such as algebra to geometry and statistics to algebra, and to bind mathematics toother disciplines such as social studies and science. Hence, an awareness of the needfor mathematics is embedded in curricular emphases and students seldom ask'When will I ever use this?'.

Third, students are expected to read the mathematics text. This is an especiallyimportant feature. Because METRO is a supplementary programme, it strives tohelp the participants develop the ability to become independent learners so thatthey can transfer learning from this programme back to the formal educationenvironment. Thus, there are also assignments and periodical evaluations to ensurethat students are learning and keeping pace with instruction.

The programme is staffed entirely by female teachers who have highexpectations of participants and are convinced that all of the girls can and shouldlearn a good deal of mathematics. for 'expectations of success often beget success'(N. jakucyn, private communication). Teachers are expected to encourage anattitude towards mathematics that is contagious. The importance of attitudes inlearning subject matter wa, addressed by Charles Merideth at a conference onmathematics for minorities:

1011.1r slIcie\S with subject matter is determined by our attitude and the

The METRO Achievement Program

attitude of those we are teaching. Attitudes have their owr distinctivearomas; students can smell them ... They know when we care. Theyknow when we believe in them, they know when we don't believe inthem, and, in fact, they take their attitudes from our attitudes. (Merideth,1990, p. 61)

Evaluation

From 1985 until 1988, only anecdotal evidence was obtained concerning theprogranmie's benefits to participants. During the 1988 summer programme withthe 12-13-years-old age group, a more formal evaluation was conducted. Girls andtheir parents were surveyed to determine the impact of the programme. Only aportion of the data is presented here: more detailed information about theevaluation can be found in Thompson (1989).

Thirty-eight parents completed the survey When asked to comment onchanges in their daughters thr they would attribute to participation in METRO,about one-third of the parents indicated that their daughter was more determinedand inure sure of herself, and one-quarter indicated that she was better able toexpress herself. All parents indicated that they would recommend the programmeto others.

Sixty girls responded to the survey. When asked about the influence ofMETROthout one-tifth claimed that they had learned to have faith inthemselves, 12 per cent commented that the programme helped them with theirstudies, and 10 per cent indicated that it was a good learning experience. The13-year-olds felt that they were given insights on high-school and courses they

should take. Participants were also asked to indicate whether the METROProgram had changed their attitudes towards mathematics. Almost half (45 percent) of the respondents said that they had discovered that mathematics could befun or that they could learn it. A tenth of the students claimed already to likemathematics when they joined the programme and reported no change in theirattitudes.

During that summer, seven high-school students who were graduates of thesummer programme were assisting as counsellors and teacher aides. Although awlect group, these students were interviewed about their METRO experience andasked to reflect upon the nnpact the programme might have had on their lives. Sixof the seven indicated that METRO had taught them they could achieve and bewhatever they wanted to be. Four of the seven indicated that they had notoriginally planned to take mathematics and science throughout high-school, butth it METRO had encouraged them to take more mathematics. Thus, for thesegirls the programme had the effect of influencing their acadennc preparation inhigh -schoolmd th,m t. preparation undoubtedly has increased the variety of optionsthey will now he able to pursue.

In an effort to obtain more current data about the impact of the programme,a questionnaire surveying a wide range of issues was administered during the spring

I13 /

Denisse R. Thompson

Table 3.1: Influence of METRO on perceptions of mathematics (as a per cent of totalresponses)

Middle-school

Agree Disagree(r7 361

High-school

Agree Disagree= 12)

METRO has increased myinterest in mathematics.

56 22 50 8

Much of the mathematicsmaterial at METRO hasbeen new to me.

53 33 25 67

Before attending METRO, 36 33 50 25

I was good at mathemans.

METRO helped me realize 50 33 25 8

I could do algebra.

Because of METRO. I plan(am taking) college-prepmathematics courses inhigh-school.

33 17 33 8

of 1992 to a sample of the students participating in the academic-year session.Several of the respondents had been involved in some aspect of the summerprogramme, but others were new to METRO that academic year. The sample issmall, so results should be interpreted with care.

Girls were given several statements about the influence of METRO on theirperceptions of mathematics and were asked whether they agreed, disagreed, or hadno opinion. Table 3.1 contains the percentage of responses to several of thesestatements for girls in the middle- and high-school years. For both groups, abouthalf of the girls indicated that METRO had increased their interest in mathematics.Substantially more of the middle-school students than high-school students tbundthe mathematics material new to them, perhaps a reflection that this is typically alevel at which there is a great deal of review in the traditional US curriculum(Flanders, 1,987). Furthermore, about half of the middle-school students indicatedthat METRO helped them reali7c they could do algebra, again, likely a reflectionof the mathematics curriculum at METRO which places a great emphasis on pre-algebra, particularly patterns, use of variablesind solving equations. About a thirdof -he participants either are taking or plan to tako college-preparatory mathe-matics because of their METRO experience. Tlic se results are consistent withthose obtained front the seven graduates of the programme who served as summercounsellors.

High-school participants were asked to comment on how METRO hadcontrihuted to the development of their mathematical ability. Table 3.2 containsa sample of responses. These girls saw METRO as encouraging them and helpingthem develop confidence in their ability to pursue mathematical tasks. Whenasked to comment on how METRO had hindered the development of theirmathematical ability, there were no responses. However, one participant

32

The METRO Achievetnent Program

Table 3.2: Responses to the question, 'How has METRO helped the development of yourmathematical ability?'

METRO helped me become more interested in math because of the way thely) challengedme and showed me different ways of working out problems.

I've always beer at math but M' 1C'i has pushed me even further to put more effortinto what I was doing and helpled] me get As instead of Bs.

METRO has helped me in persevering and not giving up when it comes to hardmathematical questions.

METRO has helped to strengthen my mathematical ability.

METRO has taken the time to teach me math at my own pace.

It [METRO) has given me easy ways to solve long word problems, it has also made melook at math in a positive way.

They [METRO] helped me get interested in math.

They (METRO) have helped me feel confident I could do the math work

METRO has helped me find strategies in taking math tests and has shown me someshort-cuts to solving problems.

[METRO helped me) find different ways of solving problems and it's more understandable

It hasn't helped much

I haven't done much math in METRO. I have however worked with the student I tutor

Note. Comments are from high-school participants (ages 14-17) only.

menfioned that, although the junior METRO Program had enabled her to enrolin honours-level classes, she had been unable to remain in courses at that level.

As previously discussed, the creators of METRO had felt a need to design aprogramme solely to meet the needs of female students. In the teenage years, boysand girls begin to discover each other and it was felt that a coeducationalenvironment would detract from the overall goals of the programme. However, nosurvey had ever been undertaken to this point to determine whether theparticipants thought the programme should include males. Table 3.3 contains theresponses (agree, disagree, or no opinion) of twelve high-school participants toseveral statements about having male students included in the programme.

Most girls thought they would make as many friends, speak up as often in class,and receive as much individual attention if boys were a part of the programme.Nevertheless, half of the girls did not want to see boys included. Those who wereagainst including boys thought they would not be as comfortable or as relaxed inlearning if boys were present. Because the sample was small and consisted only ofthe high-school age group, care must be taken in extrapolating these comments tothe larger METRO group or to other age groups. Still, the comments suggest thatfor some girls, single-sex progrmnmes may be preferable. Such programmes maygive some girls a chance to shine and take leadership roles, activities they wouldbe less likely to engage in if the progranune were coeducational.

LI .5 33

Denisse R. Tiwnipson

Table 3.3. Reactions to including boys in METRO (as a per cent of total responses)

Agree Disagree

I would like to see boys allowed in the METRO program. 42 50

I would make lust as many friends at METRO if boys wereallowed in the program.

92 0

I would talk up in class lust as much if boys were allowedin the METRO program.

67 25

I would get as much individual atiention if boys wereallowed in the METRO program.

58 17

I would learn as much in my METRO classes if boys wereallowed in the program.

42 42

Boys are more disruptive in class than girls 67 17

I would have liked METRO as much if boys were allowedin the program.

58 33

I would have liked METRO better if boys were allowed inthe program.

8 42

Male teachers at METRO would care about me as muchas female teachers

42 33

Note. Percentages are based on a total of only twelve responses from high-school participants(ages 14-171

Conclusion

The METRO Achievement Program provides model for programmes designedto encourage young females to develop their full academic potential. Too often,students of average ability, especially if they are also female, are lost in the shufflein US secondary schools and are permitted to drift through school without takingany academically challenging courses. The decisions girls make between the agesof 11 and 17 years can have far-reaching consequences for their future educational.nd career opportunities. The problems are especially acute for minorities andthose from the inner-cities where drop-out rates are high. At a 1990 EducationSummit, the following goal for US education was outlined:

What our best students can achieve no w. our average students !mist beable to achieve by the turn of the century. We must work to ensure thata significant number of students from all races, ethnic groups. and incomelevels are among our top performers. (US Department of Education,19)0. p. 3)

34

The METRO Athievement Program

The METRO Program, which combines academic work with personal attentionand encouragement, provides one avenue to help students meet such a goal andaccept academic challenges.

While additional long-term evaluation is needed, that which has beenconducted to date, both formal and informal, suggests METRO has a positiveinfluence. High expectations by the staff seem to help participants developconfidence in their ability to succeed. Although all girls may not need such single-sex programmes, a substantial number of girls seem to need and thrive in suchenvironments. In striving for equity in mathematics education, it is important thatwe meet these students' needs and provide an environment in which they candevelop their full potential.

1 I wish to acknowledge the helpful conunents provided by Natalie Jakucyn of themEnto Achievement Program during the preparati,- of this chapter.

References

M.B. (1986) 'The context of education for Hispanic students: An overview',American Journal of Education, 95, pp. 26-57.

CAMPlil..1 I , PB. (1986) 'What's a nice girl like you doing in a math class?' Phi Delta Kappan,67, pp. 516-20.

Fi J.R. (1987) 'How much of the content in mathematics textbooks is newF,Arithmeth 'fi.aclwr, 35, pp. 18-23.

'Ns ii )N, WB. and Pm -Kt lk, A.E. (Eds) (1987) 11 ivi;jorte 2000: II ivk and II i)rkersfir the"liventyrst Century, Indianapolis, IN, Hudson Institute.

MAI I II MA I [CAI 51.11 ti( i s Er )l.!( Al I( iN 13( /AILI (1990) Reshaping School Mathematics:A Philosophy and Fmmemork fOr Curriculum, Washington. DC, National AcademyPress.

MA I I I iws. W (1984) 'Influences on the learning and participation of mUiorities inmathematics'. journal.for Researth in MatIkmatio Edwation. 15. pp. 84-93.

BAy, S.M. (1990) 'Education that works for minorities: An action plan for the educationof minorities', in MAI 11MM KM Sell NCI s Em 101A1 iON BoAlso, Afakiv Mathetmnioliiirk.fOr Minorities: A Compendium of Program Proceedings, Prokssional Papers, and ActionPlans, Washington, DC, Mathematical Sciences Education Board, pp. 30-3.

Mi itu); iii, C.W. (1990) 'Banquet Address, in Mathematical Sciences Education Board',Ahdeirw Mathematio IIbrk for Minoritie.,: A Compendium of Program Proceedings,

Prof.essional Papers, and Action Plam, Washington, DC, Mathematical Science. Educa-tion Board, pp. 56-65.

MI vl IL, M.R. (1989) 'Gender differences in mathemat,cs', in M.M. (Ed)Roulb.front the Fourth Mathemano A.%ses.,tnent of the National Asn.s.,ment of EducationalProgress, Reston, VA, National ("Own il of Teachers of Mathematics, pp. 149-59.

NM It /tiAl C,111 IN( II itt TI Ai I II IV, in MAI iii ;As! u \ (1989) Curriculum and Evaluanon

7

Denisse R. Thompson

Standards .kr School Mathematics, Reston, VA, National Council of Teachers ofMathematics.

NATIONAL COUNCII OF TLACHERS OF MA EHEMAIlcs (1991) Prokssional Standards .fiv

'(Codling Mathematics, Reston, VA, National Council of Teachers of Mathematics.NAHONAI R.ESEARCI I COUNCII (1989) Everybody Counts: A Report to the Nation on the

Future of Mathematics Education, Washington, DC, National Academy Press.NEI SON, B.S. (1986) 'Welcome: Perspective on the Issues', Proceedings .from the Operation

SMART Research Cogference, Indianapolis, Girls Clubs of America, Inc., pp. 3-5.Niciioi soN, H.J. (1986) 'But Not Too Much Like School: Case Studies of Relationships

between Girls Clubs and Schools in Providing Math and Science Education to GirlsAged Nine through Fourteen'. Paper prepared for the Operation SMART ResearchConference, Girls Clubs of America, Inc.

iompsoN, D.R. (1989) AfETRO Achievement Program: Summer 1988 External Evaluation

Report, ERIC Document Reproduction Service ED 317 651.THOMPSON, D.R. and JAIWCYN, N. (1993) 'Helping inner-city girls succeed: The

METRO Achievement Program', in Cui.vAs, G. and Diuscoi L , M. (Eds) Reaching All

Students with Mathematics, Reston, VA, National Council of Teachers of Mathematics.US Di PAR I Ml N I 01 Ea it it En )N (1990) National GoalsjOr Education. Washington. DC, US

Department of Education.UsISKIN, Z. (1988) 'The sequencing of applications and modelling in the University of

Chicago School Mathematics Project (U( SMP) 7- 12 Curri in Bi um. W.Bi RP,Y, J., BII In I.R, R., HUN 1 1 1 KAIsLR-MI ssMI R. G. and PROI ni. I.. (Eds)Applicatiom and Modelling in Learning and TCaching Mathematics, Chichester. EllisNorwood. pp. 176-81.

USISKIN, Z. (1991) 'Building mathematics curricula with applications and modelling'. inNiss, M., Bi UM, W . and HuN u I Y, I. (Eds) (Caching of Mathematical Modelling andApplicatim. Chichester, Ellis Horwood, pp. 30-45.

6

Chapter 4

Who Wants to Feel Stupid All of theTime?

Olive Fullerton

Recently I found myself waiting in a long queue in a university library. When Ifinally stood opposite the student who was performing check-out service, theyoung wonun stopped and stared for several moments at the title of the top bookin the stack I handed her. That text happened to be Buxton's (1991) Math Panic.'That was the best thing about getting out of high school', she said to me as shelooked up. 'What was?' I asked. 'Never having to be involved with math again.Ever!' I asked about the source of her dislike for mathematics. 'Who wants to feelstupid a// of the time?' she responded.

I teach a mathematics methodology course for university students preparingto teach elementary school children. In her distaste for the discipline ofmathematics, the young woman perfbrming library check-out duty exemplifiesmany of the student teachers with whom I come in contact. In f'act, over the pastten years, my unofficial records of the attitudes towards mathematics of theseemerging teachers show that, in every year at least 60 per cent, and sometimes asmany as 70 per cent, of them express sonic degree of discomfort when it comesto teaching mathematics. Because of the very high proportion of females in myclasses (in my most recent class all of the students were female), this means that,as Countryman (1992) has also observed, gender remains a thorny issue in themathematics classroom.

One of the most troubling features of dysfunctional attitudes towardmathematics is loss of self-esteem. Embedded in the comment, 'Who wants to feelstupid all of the time?', are feelings of helplessness and despair. While it must beacknowledged that some gains have been made in helping women feel moreconfident and comfortable in the field of mathematics, we are still a long way fromremoving the well-established classroom barriers and practices, as well as theso( ietal biases, which make it impossible for female students to function on anequal footing with their male colleagues.

Since the phenomenon of low confidence continued to be evident year afteryear among the female teacher-candidate population. I decided to carry out astudy aimed at exploring how such low self-esteem ...volves in this otherwise

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Olive Fulkrton

confident and competent group of young women. I wanted to discover whether

it would be possible to alter the attitudes of individual students, and whether achange in teaching practices would enhance their mathematics-related self-confidence. Oyer a period of two months. I met regularly with six young womenwho volunteered to be involved in my study and who identified themselves ashaving low self-concept in their ability to 'do' and teach mathematics.

Our first meeting took an unexpected turn when the participants expressed

extreme concern over what would be done with the information I would gleanfrom the meetings. In the discussion which followed, their fears became veryobvious, culminating in their refusal to allow me to tape or even to make notesof our conversation during the meetings. Heather summed up the feelings of the

group when she said, 'No school board would hire me if it were known that I can'tdo fractions at all or that I don't understand any geometry. What parents wouldallow their children to be in iny classroom? They'd have to be crazy. Their kidswould turn out just like me and that would be bad.' I was however given completeliberty to make notes after our meetings and to use the written reflections which

they gave me at the conclusion of each get-together. Apparently, they felt incontrol of what they recorded themselves. These reflections and my own notesformed the data for my study.

At this first meeting it also became clear that, for any confidence-building tobe possible, we would have to narrow our goals and focus on a small area ofmathematics. After much deliberation, there was general agreement that3-dimensional geometry was a particularly troubling area for each student and sothis was the strand of mathematics on which we decided to concentrate. Theparticipants spent the final hour of the session writing. They recorded their feelingsrelated to mathematics in general and wrote in particular about any incidents theycould recall which they believed nuy have affected their attitudes toward

mathematics. Over and over again their statementc told how theirself-concept had

been diminished by thoughtless comments or inappropriate behaviours. For

example. Julie wrote:

I feel my negative feelings towards mathematics began when I was in thefourth grade 19 years of age]. I rementher I was learning fractions and Iwas finding the exercises in the text book very difficult. Many times Istayed after school and I received extra help from my teacher. I think itwas around the third or fourth day of the extra help when my teacher saidto me, 'You're hopeless. You'll never understand this. It's just a waste ofmy time.' From that time on. I have always con.,idered myself to be stupidin mathematics.... I expected it [thilure] to happen and I wasn't surprisedor shocked. I learned to accept my inabihty to understand mathematics.

The other participants had similar stories to tell. Their comments indicated aprofound Lick of self-contidence and poor self-esteem related to understanding.

and the 'doing' and teaching of, mathematics.As a group. we greed on a definite procedure for each meeting. Each week,

I would distribute an article which I felt was relevant to the area of geometry

.38

Who Wmts to Feel Stupid All of the Time?

identified. The participants would read the article and respond to it in writingrecording their feelings, any 'new' mathematics they learned as they read, anymemories the reading stimulated, questions which came to mind, and so on. Inour meetings, we would discuss any feelings induced by the article, and clarify anyproblems which emerged from reading their written comments. At the conclusionof each meeting, a few minutes would be devoted to having participants record anyobservations they wanted to make.

Over the course of our meetings, four issues pertaining in particular to genderand mathematics came to light. The first issue was that the participants didn'tknow how to 'talk math'; not one of them had ever been engaged in any form ofmathematical discussion. Males had dominated the classroom mathematicsdiscourse. The second issue was that they did not know the appropriatemathematics 'register' (Halliday, 1978) needed to carry on a normal mathematicsdiscussion. Third, beyond basic arithmetic skills not one of these students couldsee any relevance for exploring the discipline of mathematics. Last, the issue ofconfidence was one which permeated all of our work; all of these young womenfelt 'stupid', had feelings of incompetence reinforced by teachers, male siblings,parents and members of their conmiunities, and felt certain that they were beyondhelp as far as learning mathematics was concerned. While these issues are definitelyconnected, for purposes of clarity I will separate them and make recommendationspertaining to each.

Talking Mathematics

Our second meeting was a great revelation for me. The students had faithfully readthe assigned article and made extensive notes which they guarded rathersecretively. When it became apparent that their jottings were all similar in nature,they gradually relaxed and a much more open atmosphere prevailed in subsequentgatherings. At first we had difficulty, establishing a natural flow of conversation.When I asked them to comment on the article, there was silence. It soon becameevident that these young women were afraid of 'talking mathematics'. The fact thatI was asking them to express ideas, opinions, concerns and problems was totallyforeign to them. This, then, was our first hurdle; to become comfortable with theidea of talking in a situation where mathematics was being discussed. Beforediscussing the article, they found it necessary first to discuss their experiences inmathematics classes. Talk was unknown to them in that setting. Julie reported:

I've never talked about mathematics before except to say that I can't doit. I've never actually talked in a mathematics class. I wouldn't know whatto say. You know I don't have anything mathematical to say.

After similar opinions had been expressed by several group members, Heatherrecalled that,

When I was in elementary school I used to try to answer questions. I'd

3 9

Olipe Fullerton

wave my hand and wave it but the teacher didn't notice me. Lots of girlshad that happen to them. I guess maybe the teacher knew that we weren'tany good. We used to talk about that. Once all of my friends [girls]decided we wouldn't put up our hands at all to see if the teacher wouldask us questions then. She didn't even notice that we had stoppedparticipating.

In her written response to the first article, Jill commented:

I agree strongly with the author's suggestions that children should discussand even argue about their findings as they explore shapes. I will alwaysremember a day in grade 12 when I started crying in mathematics classbecause I couldn't understand what we were doing. A friend who hadbeen helping me a lot throughout the year moved his desk next to mine,asked what I didn't understand and proceeded to explain the concepts tome. Usually when we talked our teacher just told us to be quiet but thatday he sent both of us to the office for talking in class and the next daymy friend had to sit in the seat farthest from mine. Reading the articlebrought this memory back to me and at the same time opened up a wholenew way of thinking about talking and discussing! ... Who knows?Maybe if I had been allowed to talk I would have walked out of math classthat day understanding instead of spending my time in the Principal'soffice.

These student responses provide concrete illustrations of research findingswith respect to the significance of language in the learning process. Take, forexample, Vygotsky (1978):

Our experiments demonstrate two important facts:(1) A child's speech is as important as the role of action in attaining thegoal. Children not only speak about what they are doing; their speech andaction are part of one and the same psychological fitnction, directed toward thesolution of the problem at hand.(2) The more complex the action demanded by the situation and the lessdirect the solution, the greater the importance played by speech in theoperation as a whole. Sometimes speech becomes of such vital impor-tance that, if not permitted to use it, young children cannot accomplishthe given task.These ,lbservations lead me to the conclusion that children solve practicaltasks with the help of their speech, (IS leen as their eyes and hands. (Vygotsky,1978, pp. 25-()

The idea that language plays a significant role in learning is not a new one. Dewey(1916) wrote about the 'importance of language in gaining knowledge' and issuedthe caveat that 'education is not an affair of "telling- and being told but is an activeand constructive process involving appropriate language with direct experience'(p. 17). More recently Barnes (1971) indicated that the language of the classroom

40

Who Wants to Feel Stupid All qf the Time?

must be viewed as more than the medium of classroom communication; languagemust be seen to be part of the very learning process.

While there is extensive research and, writing on the close connectionbetween language and learning, the more specific question of the relationshipbetween language use and learning mathematics has received less attention. Thereis, however, a growing body of research which makes evident that, to some extent,mathematics learning is also dependent on learners being given opportunities to'talk mathematics'. Cockcroft (1982), in asserting that 'The ability to "say whatyou mean and mean what you say" should be one of the outcomes of goodmathematics teaching', emphasizes the significance of group discussion duringmathematics learning. Bishop (1985) contends that there is far too littleopportunity for students to talk about and share mathematical ideas. Traditionallylearners are silent in mathematics classrooms. Teachers talk and children listen orrespond in one-word phrases. Why is this significant? Bishop (1985) believes that,'A new idea will be meaningful for a pupil to the extent to which it connects wellwith the learner's existing ideas and meanings' (Ibid., p. 27). In c her words,mathematics classes must become places where mathematics is discussed. Bishopurges teachers to make way for a dramatic increase in the amount of classroommathematics 'talk'. Hoyles (1985), while stressing thL importance of learnerlearner discourse as opposed to teacherlearner discourse, also points out thatbecoming an active listener in mathematics is equally important and conjecturesthat, in the 'silences' of discussion, real learning is taking place, 'Listening forlearning is not passive; it is an active attempt to incorporate another's schemes intoone's own; it stimulates one to go outside oneself and look again at the arguments'.(Ibid., p. 207).

The mesc..s.ge of these studies is consistent and clear. Ideal learning conditionsfor nuthematics include opportunities to discuss the concepts under consideration.Yet not one of these young women recalls having 'talked mathematics' at any pointin elementary or secondary school. The one student who did attempt to discussa problem situation with her friend found herself humiliated by being sent to thePrincipal's office. Are the learning conditions experienced by these emergingteachers anomalous, or are they representative of learning conditions generallyfound in classrooms? In a Ministry of Education for Ontario (1990) review ofgrade 6 (11-12-years-old) achievement in arithmetic, geometry and problem-solving, the preponderance of children reported that they most frequently workedon mathematics alone and that their teachers most frequently worked ordemonstrated on the blackboard. From this, one can infer that 'talking mathe-matics' has low priority in most Ontario grade 6 classrooms. Similarly. O'Brienand Richardson (1987) state that they found little acceptance, on the part ofpractising teachers, that language use is an integral part of learning mathematics.Destbrges (1989) reports that, 'Fruitful discussions are rarely seen in primaryclassrooms. This is so even in the classrooms of teachers who recognize andendorse their value. When discussions are seen, they are mainly teacherdominated, brief and quickly routinized' (p. 150).

While the women in the research project didn't recall talking in their

e

4 /

Olive Fullerton

mathematics classes, they did remember that the males in their classrooms hadmany more opportunities to respond to mathematics questions posed by theirteachers than they did. This pattern of classroom interaction in which males arepreferred is well documented (Sadker, 1984; Coulter, 1993; Best, 1983). Fullerton(1993) reports that when a male in an elementary classroom was asked why hethought his teacher asked him more questions than the girls, he replied, 'Sheknows I know the answers and she knows that they don't.'

Mathematics Register

There are many situations in our lives from which special language patternsemerge. This expected and accepttd manner of expressing ideas in a particularsituation is often referred to as 'register' (Cazden, 1988; Halliday, 1978; Pimm,1987). Halliday (1978), the first to use the v.sord 'register' to refer to thesespecialized language patterns, characterizes the term in this way: 'A "register" isa set of meanings that is appropriate to a particular function of language, togetherwith the words and structures which express those meanings' (p. 65). One well-studied and documented 'register' is 'baby talk' (Cazden, 1988), the speciallanguage used by mothers when they communicate with their infants and toddlers.As well, Cazden (1988) designates as a 'register' the specialized language ofparticular professions the way doctors, lawyers. sports announcers, or automechanics talk when they are 'on the job'. Thus there is a medical register, a sportsregister, and so on. Access to appropriate registers and a degree of comfort incomprehending and using tnem can be extremely significant to those wishing toparticipate in, or understand, the relevant activities in particular situations.Mathematical language is also a 'register'. Pimm (1987) points out that 'It is notjust the use of technical terms which sound like jargon to the non-speaker (ofmathematics), but certain phrases and even certain modes of arguing thatconstitute register' (p. 76). Further, he insists that acquiring control over themathematics register is a significant part of becoming a competent mathematician

at any level.During our meetings, it be.ame clear to me that one of the conditions

preventing these six teacher candidates from discussing mathematics and frombecoming 'competent mathematicians' was their lack of ficility in using theessential mathematics 'register'. As reported earlier, when the idea of discussingmathematical concepts was introduced to the group, Jill indicated 'I wouldn'tknow what to say you know I don't have anything mathematical to say.'Heather even reported that 'When I try to say them, mathematical words just stickin my throat you know they just won't conic out like regular words.' Inother words, these young women were reporting that the language necessary forengaging in mathematical discussion had, to date, eluded them. In their writtenresponses to the weekly readings, comments regarding the difficulty of thelanguage or 'register' were common. Patricia's comments reflected an awarenessof the n,.ed for a special mathematics register: 'The vocabulary lot- the article] was

12

Who 14/ants to Feel Stupid All of the Time?

slightly mind boggling; I compared it [the vocabulary] to the psychologyvocabulary I'm learning and found that perhaps any discipline requires its ownspecific terminology and this wasn't a "bad thing" in itself.' Compare thisstatement to one she wrote several sessions earlier:

Dodecahedron, icosahedron, ... what vocabulary! How did these labelsdevelop? Why do they have to be so complicated sounding? I can neverremember all these elaborate labels! ... Another point of dread arrivedwhen this statement was made: simple quadrilaterals, for which notwo sides intersect in any point other than an end point of both sides ...'What's so scary about that, you may well ask. I DON'T KNOW. I justhate the way it is all put together.... I really start to feel out of myterritory.

When we talked about what, in fact, was 'so scary', all of the participants agreedthat their fear was likely connected to not having actually voiced any mathematicalwords out loud. Julie reported:

When I said 'the quadrilateral' today in our discussion, it was the first timeI'd ever said it. Now that I've said it TWICE I think I can make it partof my everyday language. But last night when I was reading the article andcame across the word for the first time in years, I found that I wanted tolook away ... it was as if I didn't even want to sec it, let alone say it ...it was almost as if it was obscene.

Making Mathematics Relevant

In recent years there has been a renewed emphasis on presenting the essentialinformation of various disciplines through exposure to 'real' situations. Inmathematics, more often than not, this has been accomplished through suchgimmicks as inserting children's names in otherwise artificial problems or byhaving children create their own problems. The purpose of making these so-calledconnections is to enable students to 'see' the significance of their hard work .and,as a result, achieve greater satisfitction from knowing that they can employ thosenewly acquired skills in 'real world' situations. My criticism of this approachshould not be taken as suggesting that schools should teach only mathematics thatis needed in for who can predict with any certainty what might be useful inthe future. Even if crystal-balling were reliable, I would not advocate thatmathematics be reduced to a utilitarian discipline. Rather, I am suggesting that itmight be possible and desirable for the mathematics of elenwntary and high-schoolto provide starting points in life situations familiar to studentsind for learners tobe assisted in fitting new mathematics into already established patterns of thought

in learning to make connections with what is already known.One of the barriers to understanding mentioned frequently by the teacher

candidates working with me, was that, for them, mathematics was neither relevant

43

/

Olive Fullerton

nor meaningful. They did not appreciate that mathematics permeated their lives,that their every action was in some way connected to mathematics, that the beautyand harmony of their world was due in large measure to mathematics.Countryman (1992) writes that, 'More often than not they [women] see amathematics that contains neither women nor the concerns of women' (p. 75).Laura wrote passionately about the fact that mathematics did not 'connect' to anypart of her life nor to any knowledge already within her grasp.

I needed to know why ... we were learning mathematics ... More oftenthan not I found math to be illogical and useless; what was the point oflearning about a right-angled triangle? Was I going to bump into onewhile walking my dog down the street?

Once the participants realized that they could 'read' mathematics and began to feelslightly more comfortable with geometric concepts, their attitudes changed andthey wrote positively about their new understandings. Patricia's comments aretypical:

Overall, I enjoyed discovering that math ... can be applied to everydaylife. I especially enjoyed seeing the picture of the child involved with thcmodel of the bridge ... I had never known about the rigidity of thetriangle, nor the strength of the cylinder ... This is the way math shouldbe presented.

In discussion, Doreen added, 'At least now I can connect this mathematics[elementary geometry] with things I know already and with my way of learningthings. I don't need to be "drilled- on this interesting stuff. I seem to understandit and remember it naturally.' In a recent article, Joan Countryman (1992) asksseveral important questions related to relevance:

What would high school mathematics be if women were central to ourteaching and thought? How would the metaphors change? Would we stillspeak of 'mastery', 'drill'ind 'attacking problems'? Would the emphasisstill be logic, abstraction, rationality, and right answers, or would itbecome construction, exploration, intuition? Would it shift from productto process? From exclusion to inclusion? (Countryman, 1992, p. 73)

These young women seem to have provided partial answers to at least sonic ofthese significant questions. Clearly, they felt the need to learn in more 'natural',exploratory, constructivist ways.

The Confidence Factor

Fhere was one recurring theme in all our discussions and in every writtenresponse; the teacher candidates spoke and wrote about how, in relation tomathematics, their self-worth had become diminished over the years, howparticular teachers had shattered their self-confidence. and how lack of success had

44

titaut3 to Feel Stupid All of the Tune2

destroyed their self-esteem. Their comments were filled with negative predictionsabout the success they expected to achieve in the group and their ability to helpchildren be successful in learning mathematics. They spoke openly about the factthat they did riot ever expect to be other than incompetent even with the mostelementary mathematics. Doreen seemed to speak for the group when she said,I'm only telling you this at the beginning so that you won't get your hopes upthat I'll learn sonic math and then be disappointed when nothing changes for me.Likely these others can learn but not me.'

Following the reading of the first article, we talked about the conceptspresented for more than two hours during which I gradually introduced themathematics register that was essential for the discussion. Not once did theintensity of the discussion fade or the focus shift. A well, to reinforce the ideas,we spent time examining appropriate concrete materials models of simplegeometric solids. Each participant agreed to ask questions 'no matter how dumb'.I tried to be encouraging and gave each student every opportunity to respond toone another's questions whenever possible. At the ( onclusion of that session eachparucipant had, what she believed to be, a thorough understanding of theelementary concepts introduced in the article and a grasp of the mathematicsregister essential for discussion of those concepts. Each participant had selected asolid, des( ribed it using appropriate mathematics register, suggested where she hadencoto:tered models of the solid in the 'real' world and indicated how her concernshad beeli alleviated and in what ways she still felt apprehensive about this particulararea of mathematics. Following this particular session, Jill wrote, 'It is a small stepbut it feels like quite an accomplishment .. I feel that my knowledge is expanding,shcwly but surely I feel that the very small mental framework I have for geometryis growing to accommodate a whole new wealth of information.' In a similar vein,)(qeen indicated that 'I know this increase in knowledge will positively affect myontidence when tea( hing geometry to my class.' In the oral discussion, there was

even a (. hange of tone in their voices. They sounded more confident, moreoptimistic 'Maybe I can learn this stuffafter all'. Julie commented.

Recommendations

What can we learn from this study ablut how the teaching and learning ofmathematics affects female students? I believe it shows that we have to work ona number of areas concurrently First, we must make teachers more aware of thesigniticain e of language in zhe learning process and particularly of the importanceof language in the learning of mathematics. Mathematics learning experiencesshould become 'more hke messy conversations than like synoptic logicalpresentations of conclusions' (Lampert, 1992, p. 307). Further, as Desforges (1989)ha \ suggested 'If dist ussions are to be standard practice, then quite clearly the timewill have to be made to conduct them in an undistracted fashion. This entailsreducing drastically the breadth of the curriculum that tvachers feel obliged to rushosen (p. 15(l). As w ell, teachers must be helped to realize that all learners in the

45

Olive Fullerton

classroom, not just males, must be provided opportunities for participating inmathematics discourse.

Closely connected to mathematics discussion is the issue of mathematicsrqnster. Learners of both genders must be given opportunities to use the registerof the mathematics being discussed. Since learners often require demonstration.guidance and direction from 'experts' before they can function independently,teachers must 'talk mathematics' so that learners can do the same. Once they arefamiliar with the 'register', students must then be provided with many occasionswhich require them to use the appropriate mathematical language. But let meemphasize again, females must have equal access to the concepts of mathematicsand the power and understanding that talking about them brings.

The third message extracted from the weekly study sessions is that relevanceis of particular importance to girls and young women. Several years ago. I

interviewed sixty children as part of a research project which attempted to assessthe 'condition' of mathematics education in Ontario schools. When I asked wherethe students would use the mathematics just taught by their teachers, the followingwas a typical response: `Oh, you wouldn't use that out there Miss [in thecommunity], you only use that in here [the classroom]: The young women whomet with me must have had similar classroom experiences, for not one of themrecognized the relevance of mathematics in their lives. How can mathematics bemade more relevant? Resnick (1987) discusses and supports the view that the focusof school mathematics must move away from 'symbols correctly manipulated' and'decontextualized skills ... divorced from experience'. Schoenfeld (1988) remindsus that:

Whether or not it is intended, the students' sense of 'what mathematicsis really all about' is shaped by the culture (context) of school mathematics

the environment in which they learn these facts and procedures. Inturn, the sense of what mathematics is really all about determines how (ifat all) the students use the mathematics they have learned.... [M]oreover,... the facts and procedures students learn in mathematical instructionshould he a means to an end rather than an end in themselves.(Schoenfeld, 1988, p. 82)

If we follow this reasoning, we have serious work to do in redesigning ourcurriculum and in helping teachers change their classroom roles. Typically, in mostjurisdk tions, mathematics curriculum design has been about creating lists ofconcepts which (hopefully) match children's developmental levels. Instead,mathematics (at the levels being discussed here) needs to become contextuallybased so that meaning becomes part of the exploration. I am not suggesting thatcurrent 'school mathematics' be replaced with personalized routines or 'streetmathmatics', hut I am recommending that we bring meaning to mathematics lwallowing learners see its power and beauty in everyday situations. We must makemathematics meaningful by having learners discover that it connects coherentlyacross home, school and community. Countryman (1992) puts it succinctly: 'So,

46

1/111to Wants to Feel Stupid All of the Time?

here is mathematics challenging us to see ourselves and others in connection withthe world' (p. 82).

Finally, we must be concerned about building students' confidence. Theseyoung women felt that they could not 'do' or even learn how to 'do' mathematics.Countryman (1992) asks, 'Is there something wrong with women or is theresomething wrong with mathematics?' (p. 77).. I'd like to suggest that there isnothing wrong with either; instead the problem lies with the low expectations ofsignificant others in women's lives. The expectation of many teachers, parents, andthe general public is that women will be less successful in studying mathematicsthan their male counterparts. These negative expectations are soon 'picked up' andadopted by sensitive learners. In discussing competence in mathematics, Walk-erdine (1989) comments that, 'Girls are still considered lacking when they pertbrmwell and boys are still taken to possess something even when they perform poorly.'Educators need continually to affirm women's ability in mathematics. Burton(1989) points out that:

It is unreasonable to expect that teacher input or school environment caneffect radical changes in the image of women and of mathematics insociety. At the same time it is unreasonable to deny that, as a socialinstitution, the school has considerable influence on the attitudes andbehaviours of those within it (Burton, 1989, p. 185).

In conclusion, the impact of gender in learning mathematics remains an issuefor concern. This study identified four factors which affect the mathematicslearning of female students. By addressing each of these factors. I discoveredanswers to the questions posed at the outset of this study. Low self-esteem, whichcan have a crippling effect on the ability of some women to 'do' nuthematics,evolved through the low expectations. attitudes and behaviours of teachers,parents. siblings and community members. By using a consistent and non-threatening approach, it was possible to alter the attitudes of individuahlemalestudents. By changing teaching practices in the ways described. I enabled thesewomen to make substantial gains in their ability to understand mathematics. Ofcourse, factors which affect women as they attempt to learn mathematics affect alllearners to varying degrees. 1 believe that when we understand the types ofclassroom practices which benefit females, we will have enhanced our under-standing of mathematics learning in general. And then, all learners will benefit.

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Educators. Toronto, Queen's Printer for Ontario.0.13R ii ti, M.K. and lk uii Alt1)\()N, A. (1987) '71/at's the Only Maths 'Way.' '1;lhoo!'

Beaury!': Qualitatirv ()bsorations of Grade 4 Mathematics Classes, ERIC. 299 110.

PININ1, D. (1987) Spwking Matlumatically Lcurcion. Rondedge and Kepm Paul.

Pixim. I). (1991) 'Communicating mathematically'. in DvitKiN, D. and SHIM B. (lids)

Laiwuqe m Mathematical. Education: Research and Mactiw, Buckingham. Open Uni-versity Press, pp. 17-24.

K. I,. (1987) 'Learning in school and out'. Educanow/ Rem'archer. 16, pp. 13-20.

S sick/ it, M. (1984) Failing at Fairao: How America's Schools Cheat ( ;irk New York.

Scribner..Sc icci N.11111, A.1]. (1988) 'Problem solving in context'. in CH A14.1 is. R.md Sn sti&. F.

(Eds) The leadinti! and .-1esinis., of .1 lath( tnalkal Pcobleni Solving, Reston, Nanonal

Council ofTeachers of Mathematic., pp. 82-92.Vs ccci sKy. 1..5, (1978) Albid in Society: Me neeelopment of I ligher Pqchological Pwo,es,

Cauribi idge, MA. 1 Lirvard Ulliversity Press.WA11.11tItr\t, V. (1989) counting Gi.c OHL 1.ondon, Virago.

48

Chapter 5

Assisting Women to CompleteGraduate Degrees'

Lynn Friedman

Introduction

In the past five years. those concerned with gender equity in mathematics havewitnessed n increasingly bright picture. In many western countries, programmesdesigned to appeal to women have appeared and been publicized (see reports inBurton. 1990). In the US, meta-analyses of journal and other published accountsindicate a decrease in the size of gender differences in mathematical achievementreported (Friedman. 1989; Hyde, Fennema and Lamon, 1990). A survey of newdoctorates in the US reported a gradual increase in the proportion of degreesgranted to women from 6 per cent in 1950 to 24 per cent in 1993 (AmericanMathematical Society, 1993). The first president of the AWM (Association f)rWomen in Mathematics) gave a glowing account of the growth of thatorganintion over the past twenty years (Blum, 1991).

Mathematics is an addictive occupation, or preoccupation, regardless ofgender. Becker has described the features of mathematics which draw both womenand men to the subject: 'its logical nature, its problem-solving aspects. itsobjectivity, and its creative nature' (Becker. 1990, p. 120). In the mid-1960s, whenI completed my degree in mathematics, I felt that th,s support women who weieconsidering mathematical careers needed most was company in guilt guiltoccasioned by their avoidance of personal and communal social responsibilities.But. securing company in guilt meant being able to find other female mathema-ticians, and there were fewer around then than there are t'

Undoubtedly, the situation has improved. This is (ultainly true in the westernworld where women faculty are becoming more numerous and increasingly visiblein mathematics departments. However, they are still not well-represented at thehigher levels of the profession. The proportion of women receiving graduatedegrees in mathematics is still too small. In the US, gender imbalance is no longerevident at the undergraduate level. The proportion ofwomen receiving bachelordegrees in mathematics climbed from 23 per cent in 1950 to 37 per cent in 1970and then to 47 per cent in 1992. Table 5.1 gives statistics on bachelor degreesawarded in the period 1949 to 1992.

4 9

Lynn Friedman

Table 5 1- Bachelor degrees awarded in mathematics to women. 1949-92

Year Total Women %Women

1949-50 6382 1440 23

1959-60 11399 3106 27

1969-70 27442 10265 37

1974-75 18181 7595 42

1979-80 11378 4816 42

1984-85 15146 6982 46

1989-90 14597 6785 46

1990-91 14661 6917 47

1991-92 14783 6895 47

Note. Vance Grant (private communication, 1993) provided the 1990-1 figures; Norman Brandt(private communication, 1994) provided the 1991-2 figures.Source. US Department of Education. Office of Educational Research and Improvement. (1989)Digest of Education Statistics. Twenty-fifth Edition. Washington, DC, US Government PrintingOffice, Table 266, p 281, US Department of Education, National Center for Education Statistics.(1991) Table 233

Table 5 2 Masters degrees awarded in mathematics to women, 1949-92


1949-50 974 190 201959-60 1757 335 19

1969-70 5636 1670 301974-75 4327 1422 33

1979-.80 2860 1032 36

1984-85 2882 1008 35

1989-90 3667 1472 40

1990-91 3615 1478 41

1991-92 4011 1559 39

Note Vance Grant (private communication. 19931 provided the 1990-1 figures. Not man Brandt(private communication, 1994) provided the 1991-2 figuresSource US Department of Education. Office of Educational Research and Improvement (1989)Digest of Education Statistics, Twenty-fifth Edition. Washington, DC. US Government PrintingOffice, Table 266, p 281, US Department of Education, National Center for Education Statistics(1991) Table 233

The proportion of women receiving masters degrees in mathematics hassimilarly increased (.)ver the years. As is apparent from Table 5.2. 39 per cent ofthese degrees were awarded to women in 1992.

The figures which are cause for concern, howeverire those for doctoraldegrees. Table 5.3 conveys this information and shows that the percentage ot'female recipients has increased over the last tbrty years, front 6 per cent in 1950,to 21 per cent in 1992. Furthermore, according to ,1 survey conducted by themajor mathematical organizations in the «)untry in 1993 the figure stood at 24per cent (American Mathematical Society 1993). However. even the latest figuresare not satisfactory. According to Henrion (1991), women make up less than 6 per

cent of tenured faculty in mathematics.

50

Assisting 14;iinen to Complete Graduate Degrees

Table 5 3 Doctoral degrees in mathematics awarded to women, 1949-92


1949-50 160 9 61959-60 303 18 61969-70 1236 96 81974-75 975 110 111979-80 724 100 141984-85 699 109 161989-90 915 169 181990-91 978 188 191991-92 1082 231 21

Note: Vance Grant (private communication, 1993) provided the 1990-1 figures; Norman Brandt(private communication, 1994) provided the 1991-2 figures.Source US Department of Education, Office of Educational Research and Improvement (1989)Digest of Education Statistics, Twenty-fifth Eth'ion. Washington. DC, US Government PrintingOffice, Table 266, p 281, US Department of Education, National Center for Education Statistics.(1991)Table 233.

Table 5.4 SAT-M effect si7e5 and ratios of males to females in the upper score ranges, 1981-90

YearEffect Size

OverallRatio

Upper 0.05Ratio

Upper 0.01Ratio

Upper 0.005

1981 0 43 2 67 3 40 7.411982 0 44 2 76 3 58 8.141983 0 41 2 57 3 29 6 991984 0 39 2 48 3 12 6 571985 0 40 2 48 3 11 6.311986 0 42 2 64 3 40 7 471987 0 39 2 55 3 28 7 261988 0 37 2 33 2 90 5 751989 0 39 2 53 3 25 7 171990 0 37 2 38 2 99 6 19

Note Data for this table was calculated from numbers of students taking the SATM, meansand standard deviations, as reported by the College Entrance Examination Board ResearchDepartment, in private communications, dated 1989, 1991, and 1992

The rise in the number of mathematics degrees granted to women is surely causetbr optimism. However, researchers dealing with highly selected elite populations,such as gifted junior and college-bound senior high-school students, pointto figureswhich are not quite so encouraging. For munple, for Stanley and Benbow's'mathematically precocious youth'. the ratio of males to females is large. Theseindividuals were tested using the mathematics section of the SAT-M (ScholasticAptitude Test)i US college entrance examination on which scores can go as high as800. For these gifted samples. the higher the group cut -offscore, the larger the ratioof males tofemales, reaching 12 to 1 for scores over 7110 (Stanley and Benbow, 1982).These figures are similar to those for college-bound seniors (i.e., final yearsecondary-school students planning to attend college). Table 5.4 gives statistics forSAT- M scores from 1981 to 1990 and includes the ratio of males to females in each of

Lynn Friedman

the 0.05, 0.01, and 0.005 upper score ranges of the distributions.Ratios in the upper score ranges do not reflect the effect sizes (that is, the overall

standardized mean differences) between males and females on the mathematicsportion of the SAT. These have recently decreased to slightly more than one-third apooled standard deviation. However, SAT-M male variances are substantially higherthan female variances. In this case, a small mean difference, even one in favour offemales, can lead to large male to female ratios at the upper ends of the distributions(see Becker and Hedges, 1988). It is true that since the mid-1970s, the ratios ofStanley and Benbow's (1982) gifted population in the upper score ranges have

decreased. Moreover, the percentage of women in many other mathematicallydemanding professions such as computer science, engineering, and the physicalsciences, is smaller than that in pure mathematics. However, those of us who areconcerned with gender equity may take little consolation in this.

Opinion is divided on whether women entering doctoral programmes in the USdrop out at higher rates than do men. Lewis (1991), while noting that concrete data isdifficult to obtain, states that .. women constitute only about 30 per cent of thosepursuing a curriculum that leads directly to a doctoral programme. Thus, the fact thatwomen constitute about 25 per cent of US citizens earning a mathematics doctoratewould indicate we are not losing many well qualified women at the doctoral level andthat to increase the number of women doctorates requires getting them intoappropriate undergraduate programs' (Lewis, 1991, p. 722). Other writers disagree(for example. Billard. 1991; Case and Blackwelder, 1992). Most of the evidence citedis anecdotal, although Billard reports a 1983 study, conducted by Berg and Ferber atthe University of Illinois. which found a higher attrition rate for women despitegender-equitable distribution of financial aid. Research dealing with this question isin progress (see Case and Blackwelder, 1992).

This, then, is the problem: women receive small proportion of the highestdegrees awarded in mathematicsmd possibly drop out of graduate programmesat higher rates than do men. The question therefore becomes, what does theresearch tell us about how to attract and retain women in these programmes?Qualitative research, in the form of interviews and case studies, has provided manysuggestions. Sonic of the findings, in fact, are reminiscent of those confirmed inlarge studies of younger students. Helson, cited in Becker (1990), found thatwomen mathematicians were less assertive, less confident, and less comfortable intheir social surroundings than were their male counterparts. Becker. herself,observed that female graduate students had more doubts about attending graduateschool than did men. Confidence in abilitvissurance in pursuing an advanceddegree. and comfort in the community of mathematicians are clearly areas thatmust be encouraged.

Confidence and Preparation, Interacting Factors

The finding documented most repeatedly in studies of Younger students is thatwomen have less confidence in their mathematical skills than do men (for example,

Assisting 11.'omen to Complete Graduate agrees

Benbow, 1988; Sherman and Fennema, 1977). In fact, confidence is a centralfactor in women's participation in mathematics at all levels beyond elementaryschool (Armstrong, 1985). Recent reports indicate that women sire still participat-ing in the most advanced courses in lower numbers than menilthough there areindications that the situation is changing (Frazier-Kouassi et al., 1992).

Weak preparation for university compounds the problem. Researchers havefound that women tend to attribute failures to internal causes and success toexternal ones, while men do the opposite (see, for example, the discussion inKloosterman, 1990). As such, women may blame their abilities for problemswhich are caused by their inadequate preparation. Worse still, according toHarrison (1991). 'peers and professors probably will make the same judgments dueto the same social pressures' (p. 732). These circumstances can only function todiminish confidence.

Regardless of their preparation, women do comprise nearly half of theundergraduate students in mathematics. However, confidence may once again bea factor in their seemingly less ambitious choices of focus within the field (Frazier-Kouassi et a/., 1992). Furthermore, women graduate students often make theunfortunate decision of beginning in a masters programme to see how they likegraduate studies instead of enrolling directly in a doctoral programme, not realizingthat by doing so they may be labelled as 'not serious about doing mathematicsresearch (Harrison, 1991).

It is possible that encouragement from teachers and other experts in the fieldoffset lack of confidence. In Becker's study, 'the women were more

influenced by "significant others- than were the men (or at least they were willingto admit it)' (Becker, 1990, p. 126). However, women are apparently less likely tofind encouragement. In their undergraduate samples. Frazier-Kouassi et a/. (1992)observed that males received encouragement more often than females to takechallenging mathematics and physics courses. However, persistent personalencouragement also has its limitations. Seymour (1992) cautions that women, whotend to 'perform for others' inure pervasively than do men, may be less able tohandle the transfer to a more impersonal environment.

Nonetheless, sensitive and accessible advising is crucial. The problem is it isnot always available. '... [Wlomen and minorities are frequently niadequatelycounselled, and ... ilanything. they require more counselling than other studentsto combat the gender and racial-stereot-yping they receive.... [Title need forbetter advising at the post-secondary level has become .1 serious problem' (Keith.1989 p. 21). Moreover, mathematics faculty ti-equentiv believe that the onlyads ice A graduate student needs Is from the thesis advisor. 'For advising to besut Lessful, its importance must be recognized' (National Research Council,United States. 1992, p. 35).

Counselling and help witn academic problems should be accessible to allstudents on A regular and frequent basis. In addition, it should be available frommathematics ficulty within mathematics departments. Women's groups or specialAi( es created to aid women and minorities ACC not adequately equipped to guideand support women.in speciahied academic programmes.

5.3

400

Lynn Friedman

Advisers and counsellors must be nude aware of women's possible lack ofconfidence. Women must be encouraged to tackle ambitious programmes, and

they must be provided with opportunities in lower-level graduate courses to rectifyinadequate preparation. Faculty in mathematics departments must be sensitized tothe need for a well-structured and supportive advising programme catering to allgraduate students, but especially to the particular requirements of women.

The Social Environment and the 'Critical Mass'

The discomfort women experience in the social environment of nuthematicsdepartments ranges from feelings of being viewed as annoying intruders to simply

being ignored. Social interaction between graduate students may be centred onsports in which the female student has no interest. Worse, there may be noongoing social interaction at all, since faculty and advanced graduate students maybe distant and engrossed in their own research. Such isolation is clearly diminishedwhen there are other female students around. Harrison (1991) notes that lt]heseproblems [of male-female interaction] diminish when there are enough womenaround. The men are more accustomed to their presence and the women have

each other for support (p. 732). Frazier-Kouassi et al. (1992) cite evidence that'women are more likely to ... remAn in a nontraditional job or programme if there

are a reasonable number of other fenules present as classmates and/or as colleagues'(p. 49). Jadcson (1992) found that one important characteristic of departments that

are top producers of women mathematics doctorates is that they have women onthe faculty. Female liberal arts colleges provide a disproportionate number ofstudents who enter and complete graduate programmes in mathematics (Sharpe,

1992).The proportion of females that might be considered adequate to ensure

comfort in the mathematical milieu, that is provide a 'critical mass', probably variesti-om institution to institution. However, in the interests of assisting women to findgraduate programmes with A 'critical mass' of female involvement. informationshould be made available on the number of women faculty and graduate students

in each institution. ln 1976, Herstein (as described in Schafer, 1991) reported ondepartments that had produced female doctorates. More recentl); Jackson (1991)has provided an account of the top producers of women mathematics doctoratesbut. considering the time that it takes to produce a doctoral thesis, someuniversities may be doing better now than those on Jackson's list. A completeWing, hv iwitution, of US doctoral degrees granted in 1992-3 has beenpublished in the November 1993 issue of the Noneo of. the Americas, .11athemati(alSociety.

'74

Assisting iromen to Complete Graduate Dwrees

Financial Support

The western world has produced a substantial amount of legislation intended toeliminate barriers to women entering degree and professional programmes. Muchof this legislation revolves around money, either by prohibiting the use ofgovernment resources by establishments which do not comply with equitystandards, or by awarding grants directly to equity projects. However, earlyexpectations concerning the levels of funding which would be provided haverarely been metind in many cases funding levels have remained the same or evendeclined since the early 1980s (Stromquist, 1993). Indeed, 'Gender is an areaparticularly vulnerable to symbolic politics ... the rhetoric surpassed funding andimplementation' (Ibid., p. 399). Unfortunate, as a collective, women mathema-ticians do not yet have the political strength required to campaign effectively forbetter levels of support. One reason for this may be that their efforts are dispersedamong several groups within a variety of larger professional associations, and incontrast to their parent ass-ociations, there is little sharing of information andcombining of effort among them.

Delivery of Instruction

Apart from research on confidence and participation. our information is sparse inthe area of the environmental factors affecting women in mathematics pro-grammes. For the past forty years, there has been sonic focus on cognitive factorswhich interact with mathematical skill. In the late 1950s, the Catholic Universityof America produced several doctoral theses analyzing gender differences inmathematics. More specifically it was reported that spatial and verbal skills interactwith nuthematical achievement in different ways for each gender. In flit, one ofthese researchers was prompted to suggest that since girls and boys evidentlylearned mathematics in different ways, they should be taught by drawing ondifferent cognitive strengths and possibly in different classrooms (Emm, 1959).More recently. other researchers have found that females who attend single-sexschools attain better results in mathenutics than do those in coeducationalenvironments (see, for example. Lee and Bryk, 1986). However, evidence distilledfrom studies since the 1950s does not support the notion that males and femaleshave differences in cognitive learning patterns except. possibly. in academicallyelite samples (Friedman. 1994).

It has been suggested that cooperative learning environments may be inuresuccessful in nourishing wolnell's mathematical growth (for example. Barnes andCoupland, 1990; Frazier-Kouassi et al., 1992). Other research, however. indicatesthat such an approach must proceed with caution (Webb. 1984). Women in smallgroups, where they are outnumbered by men. may find their voices drowned outand tlwir confidence thus fiirther diminished. Fra/ier-Kouassi et al. recommend anemphasis on problem-solving, model-buildnigind the discovery approach toteaching. However. research supporting the effectiveness of these approaches forfemales is scant.

55

Lynn Friedman

Should we wait for the evidence? Authors of the University of Michiganproject believe that 'we cannot afford to wait for research results before beginningto implement institutional change' (Frazier-Kouassi ci al.,1992. p. 131). On theother hand, Seymour (1992) states that '... some commonly expressed, butuntested, theories about the causes of switching [out of mathenutics and scienceundergraduate majors] have limited validity, and may serve to divert attention awayfrom effective institutional interventions' (Seymour, 1992, p. 292). Apparently,doubts as to the theoretical viability and practical effectiveness of potentialinterventions was one reason for the 'cool reception' given to a group of universityresearchers by funding sources (Lewis, 1991). Of course without attempts atreform, the necessary research will never exist. Srliporting many small-scaleintervention projects may be more effective in the long run than pressing for

universal methodological change.

Conclusion

The logical, objectiveind non-verbal nature of mathematics, opening new worldsof reasoinng to those who like to use reason, has captivated most of the readers andwriters of this collection. Mathematics as it is. or waS, appeals to women.Moreover, there is the emergence of a new kind of mathematics, which may beeven more interesting and viable thr females, in the application of mathematicalmodelling to areas such as the environment and public policy. We need to reinforcethe strategies of the -dast while snnultaneously searching for new methods and wecan do both.

The women entering mathematics graduate programmes today are still .oftenambivalent or uncertain. The experience of Women today is not So ditThrent fromthe experience of women of the I 960s .1s I remeniber it women still feel al)obligation to make a contribution to society, and niany are unaware of the socialvalue of mathematics.

We must make continued efforts to encourage change in graduate mathe-matics programmes: we must press thr improvement in the collection anddissemination of information about female enrolment, and in the quality andquantity of the advising offered to women. Sensitive advising and the presence ofcritical masses of women interact to benefit fcmale graduate students byreinforcing their confidence in their own abilities, promoting their social comfort,,ind enabling them to exchange information about the social value of nutheniatics.Finally. we imist find ways to join forces by integrating rather than dissipating ourefforts through the nuny organizations concerned with gender equity inmathematics. Only then can we apply our combined strengths to achieving theseshared goals.

56

AssistivIt'omen to Complete Graduate Degrees

Note

The author is grateful to Carol Wood. former president of the Association for Womenin Mathematics of the United States. for many helpful suggestions during the time thiswork was in preparation.

References

ANti RICAN MAMFMA1 ICA! SOCIk 1 N' (1993) '1993 annual AMS-IMS-MAA survey'.Notices orate .-Imerican Mathematical Society 40, pp. 1164-99.

ARAI,' Rom:. J.M. (1985) 'A national assessment of participation and achievement ofwomen in mathematics'. in CIIIPNIAN. Bikusi c, L.R. and Wit \ON. D.M. (Eds)11;nnen and .11atheniatic: Balancing the Equation, Hillsdale. NJ, Lawrence ErlbautnAssociates. pp. 59-94.

BAILNI , M. and Coun AND, M. (1990) 'Humanizing calculus: A case study in curriculumdevelopment', in But ioN. L. (Ed) Gender and Mathematics: An International Perspectii.e.England. Cassell Educational Limited, pp. 72-9.

Bc }:1 .1.11.. (1990) 'Graduate education in the mathematical sciences: Factors influencingwomen and men'. in Bus. i. N. L. (Ed) (;ender and Alathematics: An InternationalPerspative. England. Cassell Educational Limited. pp. 119-30.

131( I:I us., BJ. and Him:i s. L.V. (1988) 'The effects of selection and variability in studiesof gender differences% Behavioral arid Brain Sciences. 11. pp. 183-4.

Bi xueov C.P. (1988) 'Sex ditThrences in mathematical reasoning abihty in intellectuallytalented preadolescents Their nature. effects. and possible auses'. Beharioral and BramSciences. 11 . pp. 169-32.

Bui k L. (1991) 'The past. present. and future of academic women in the mathematicalsciences', Notices ot. the American .1fathernatical Society 38. pp. 707-13.

BI UNI, L. (1991) 'A brief history of the Association for Women in Mathematics: Thepresidents perspectives'. Notices of the American Mathematical Society 38, pp. 738-54.

Bilk] (1... I.. (Ed) (199(i) Gender ,i/id Mathematics: .4n International Perspectire, England.Cassell Educational Limited.

CAsi B.A. and Bi ACKWII 1)1 R. M.A. (1992) 'The graduate student cohort, doctoraldepartment expectations. and teaching preparation'. Notice., of the .-Imerican Mathemat-ical Society. 39. pp. 412-18.

Emxt, M.E. (1959) .4 Factorial Study of the Problem-SA.1v Ability ef lyih-Grade Boys.Doctoral dissertation. Washington. DC. The Cathohc University of AmericaPress.

FR sin R_Kot-Assi, ti.. MNI A \ till:. 0., SI . P., BLINKAM. GrItIN, K. )1 I I

III Al N C.. LI WI's. DJ.. Si )11 I NI \ . P. Ni NI , II. and Dwis. C. (1992) 11 :mice,In Mathematics ,nol Physics: Inhibitor.. and Enhancer'. Aim Arbor. University olMichigan,Center ('or the Education of Women.

I. . (1)89) 'Mathematics and the gender gap: A meta-analysis of recent studieson sex differences in mathematical tasks'. Renew 9 Fdwational Re'ran h. 50.pp 185-213.

Fun I.. (1994) 'The role of spatial skill in gendsr difference in nuthemancs: Meta-analytic evidence', Paper presented at the annual meeting of the American EducationalResearch Society, New Orleans.

Lynn Friedman

HAlilt )r:, J. ( 1 99 1) 'The Escher Staircase', Notices of the American Mathematkal Society 38,

pp. 730-4.HENRION, C. (1991) 'Merging and emerging lives: Women in mathematics'. Notices of the

American Alathematical Society. 38, pp. 725-9.J.S., FENNI.MA, E. and LAMON, S.J. (1990) 'Gender differences in mathematics

performance: A meta-analysis', Psychological Bulletin, 106, pp. 139-55.JACKSON, A. (1991) 'Top producers of women mathematics doctorates'. Notices of the

Americai, Mathematical Society 38. pp. 715-20.Ki au, S.Z. (. 989) 'The conference on women in mathematics and the sciences', in 1(1.11

S.Z. and KEI ii i. P. (Eds) Proceedings of the National Conference on Iiinnen in Mathematksand the Sciences, St. Cloud, Minnesota. pp. 1-5.

oos I LRMAN, P. (1990) 'Attributions, performance following fiilure, and moi vation in

mathematics', in FUNNI MA, E. and LEDIA. G.C. (Eds) Alathemarks and Gender, NewYork, Teachers College Press, pp. 96-127.

LI I V.E. and BIO'K, A.S. (1986) 'Effects of single-sex secondary schools on studentachievement and attitudes',Journal of Educational Psychohwy, 78, pp. 381-95.

Li wis. DJ. (1991) 'Mathematics and women: The undergraduate school and pipeline'.Notices of the American Mathematical Society 38, pp. i21-3.

NA I I( )NAI Ri si ARCH CA )UNC:11 UNI I ) S IA I / s (1992) Lineal* rthematical Scientists:

Doctoral Study and the Postdoctoral Experience in tlw I'nited States. Washington, DC.National Academy Press. (Available from National A ..-ademy Press. 2101 ConstitutionAvenue, NW. Washington DC, 20418)

R.AMIS t. L. and ARM I I I S. (1986) Profiles, Collve-Bomid Senioo, 1985. New York,Colkge Entrance Exannnation Board.

Sci 1AI I IL, A.T. (1991) 'Mathe mat.cs and women: Perspectives and progress'. Notico 0: theA»wrican A latheinatkal Society 38. pp. 735-7.

Si VMOlflt.. E. (1992) 'Undergraduate problems with teaching and advising in SME majorsexplaining gender ditTerences in attrition rates', journal of Colhge Science li.achiv,

21, pp. 284-92.SuARPI N.R. (19921 'Pipeline from small colleges' (letter to the editor( , Notice (!ie the

Am( lean Mathemancal Society 39. p. 3.u ikm J. and Fi \\INA E. (1977) 'The study of mathematics by girls and boys: Related

variables'. American Educational Research iournal, 14. pp. 139-(t8.S; \ I I J. and BI xiii AT.; C. (1982) 'Huge ex ratios at upper end' [Comment!,

Psycholi.q. 37. p. 972.S i Rl IMQt. I i. N.P. (1993) 'Sex equity legislation in education: The state as promoter of

women's rights'. Review of Educanonal Research, 63, pp. 379-407.Wi nu, N.M. (1984) 'Sex differences in interaction and achievement in cooperanve small

groups '. low nal of Lim ational Psychology, 7 (1, pp. 33-44.

.5S

t

Chapter 6

A National Network of Women: Why,How, and For What?

Barbro Greithobn

Flus ( hapter discusses the origins and goals of the Swedish Women andNlathematics network and outlines its connections with the society within whichit operates. Through ( onferences and newsletters, women support each other,share intOrmation and collaborate on int projects. Published conferencepro( ceding, Ill turn provide the literature for pre-service and in-service teachereducation and in this wav the ideas of Women and Mathematics are more widely.'.;-:.enunated. As well, we are engaged in research on classroom interaction andtirriculum developmentind other activities which promote womm in post-

set ondarv education.

Why the Need for a National Network?

Sweden is popularly regarded as a country which has made great strides inat lueving equality between men and women. But, as we shall see, in the field ofadvanced mathematics this is still fir from being realized. The Swedish educationalsystem offers equal access to all students, providing nine years of compulsoryedut anon, and an additional optional three years at the upper-secondary schoollevel, which the vast majority of students choose to do. It is generally agreed that

perfOrmance in mathematics throughout the compulsory years of educationis better than that of boys, despite the fact that boys obtain slightly better resultsoil average than girls in the national standardized test given in year nine (Grevholmand Nilsson. 1994). This pictur- is very similar to that found in many other studiesot gender differences in mathenutics achievement (Kimball, 1989).

1 he upper- se«mdary level is where different gender patterns of participationin mathematics begin to be apparent. I.ess than one-tifth of all students choose topursue the natural science programme, which is a requirement for those intendingto go on to university to study mathematics or science. Of this group or one-third are gIrls. I loWever they are often very successful and leave upper-set ..,adaryst !tool with higher marks in nuthematics than boys. Indeed, according to Ljung

59

t)

Barbro Grevhohn

(1990), as a group, girls who choose to study natural science enter upper-secondary school with better marks than boys, and obtain better results than boyson the national standardized mathematics test taken in the third year of upper-secondary school.

At the university level, this trend continues. Fewer than one-third of allstudents in mathematics are women (although 40 per cent 'of students preparingto teach secondary-school mathematics are women). However, at the Universityof Lund, Mathematics Department statistics show that while 55 per cent of malestudents finish their programme in the expected period of time, the correspondingfigure for women is 60 per cent. Nonetheless, very few women go on to dodoctoral work in mathematics. The first woman in Sweden to obtain a doctoraldegree in mathematics was Louise Petren who received her degree from LundUniversity in 1911. Since then, Swedish universities have granted only twenty-onemathematics doctorates (or equivalent degrees) to women (Grevholm, 1994). Andaccording to Piene (1994), Sweden's record in this regard is worse than all otherScandinavian countries.

This poor record of female participation mathematics is further reflected inthe flict that only 3 per cent of senior university mathematics lecturers are womenand Fweden currently has no female professor of mathematics. Indeed, Sweden'sonly female professor of mathematics was Sonja Kovalevskaja who died in 1891. Incontrast, in other disciplines, on average about 8 per cent of all professors and 20per cent of all senior lecturers are women (Wittenmark, 1993).

Clearly, there is a need to increase female participation in mathematics bothat the upper-secondary level and at the university level. There appear to be twocrucial stages in a girl's life where major barriers to participating in mathematicsare experienced: the first is the point of choosing a programme of study in upper-secondary school, and the second is the passage to graduate studies. Evidently,systems change very slowly if they are not pressured from outside there has beenlittle change in twenty years in the field of mathematics, while in medicine andlaw the rate of participation of women has increased considerably over that periodto the point that it is now almost 50 per cent. To change the prevailing pattern inmathematics education intervention in the educational system is essential.

The Swedish network of women seeks to increase the number of females inmathematics by engaging them in various kinds of intervention. From a theoreticalpoint of view, the network is an intervention project, but underlying our work are

a variety of feminist assumptions (see Mura, Chapter 19, this volume). Accordingto Fennema (1994), personal beliefs and perceptions, as well as methodology,including the scientific method, are all influenced by male perspectives. Becausefemale perspectives have not been included in a variety of areas, our view of theworld is interpreted throuyb male eyes and is incomplete at best. The ways inwhich women are invisible and (made) passive can be seen very clearly in variousmathematics organizations of Sweden. The Swedish Mathematics Society, forexample, has very few female members and only one woman on its board.Teachers' associations in mathematics and science are male-dominated eventhough there are many female teachers in these subject areas. For instance, the

60 73

.4 National Network of Mmen

eleven-person organizing conmiittee for the 1993 Mathematikbiennalen, abiannual national conference for mathematics teachers, had only two womenmembers. And although most of the teachers at the younger age levels are female,only forty-nine of the 114 speakers were women. However, at the same time, itis encouraging to note that the situation is better now than it was up to 1990. Asanother example of the way in which women are made invisible, Swedishmathematics textbooks paint a pictwe, through their choice of illustrations andproblem contexts, of a world that is two-thirds full of men (Areskoug andGrevhohn, 1987; Rönnbiick, 1992).

Why then have we chosen to establish a network consisting only of women?Why not a network for all mathematicians and all teachers of mathematics? Do\vomen need ways to reach each other and handle problems away from men?Projects in which boys and girls are taught separately are currently underway andmembers of the network are examining the effects of such interventions(Rönnb:ick, 1992). Sonic researchers (see, for example, Hanna, 1994) have warnedabout the negative effects of segregated teaching. Undoubtedly questionsconcerning the advantages and disadvantages of segregation are worthy of deeperanalysis, but fbr the moment, at least, many women seem to value working in asegregated environment.

How Are We Organized?

There are of course many ways we might have chosen to organize ourselves inpursuit of our goal to empower women in mathematics. What are the advantagesof a network? A tbrmal organization would certainly offer opportunities forwomen to meet and make contact, but it would also require considerableorganizational effort which would divert energy from other pursuits. A networkoffers the same opportunities without the need for special formal arrangements.A network empowers women by giving them visibility, helping them to makepersonal contacts, opening up communication, and making it possible for them toexplore each other's work and results. It provides the inspiration and motivationfor women to combine their strengths and work together to achieve a commonpurpose. The aims of the Women and Mathematics network in Sweden are:

to create contacts between those who are interested in issues concerningwomen and mathematics, both in teaching and in research;to disseminate information on projects and research about women andmathematics;to suggest speakers (preferably female) on subjects concerning women andnuthematics; andto be a national section of the international 10WME network (Grevhohn,1991)

In the four years since the network was established, the membership hasgrown from 130 to about 500 personsi growth which demonstrates very clearly

6 /

Barbra Greuholin

Figure 6.1: Model of connections between the Network and the surrounding societyNote: A double arrow shows that persons belong to both organizations, a single arrow indicatesthe direction of possible influence.

National Agency for Education Swedish Parliament

Swedish Mathematical Society

SwedishNetwork

Teachers ofmathematicsand science

Mathematicsbiannual

conference

Journal forteachers

Network in Denmark Network in Norway

the need felt by women for finding ways to meet and discuss these issues.Connections are facilitated and information is disseminated by means of anewsktter which is distributed twice year. Included with the national newsletteris the Newsletter (y. the International Orvni.:,ztion of I l'omen and Matluvatics, and thisensures that news about gender and mathematics front all over the world rea,-hes

62I-

S

.4 National Network qf Iflowen

women in Sweden. Two national conferences have been organized by themembers of the network and the published conference proceedings are now usedas literature in teacher education (Grevholm, 1990; Brandell, 1994). The firstnational conference in Malmö served as a stimulus to women in the other Nordiccountries: in 1991 a conference took place in Denmark (Tingleff. 1991), and in1992 a conference in Norway was arranged (NAVF, 1993). The three networks inSweden. Denmark and Norway have several members in conunon and thisprovides excellent opportunities for conmiunication and the sharing of resourcesacross countries.

Figure 6.1 illustrates the relationship between the Women and Mathematicsnetwork and other structures and organizations of Sweden and beyond. Someexamples follow of the ways in which the work of the network has both influencedand been influenced by the various groups represented by the membership (formore examples, see Brandell. 1994 and Grevholm, 1992a, 1992b). Several of ourmembers are textbook authors. As they became aware, through the network, ofgender inequity in textbooks they were, in turn, better able to influence their maleco-authors. The first investigation, in 1987, pointed out that boys were featuredat least twice as often as girls in textbooks. An investigation in 1994 showed thata recently published textbook for upper-secondary school mathematics containsforty-three pictures of boys and men and only five of girls or women. Obviously,there is no reason to discontinue working on changing the representation of thesexes in textbooks.

Our contacts with the National Agency for Education have led to oursecuring financial support to hold our own conferences, to sponsor theparticipation of our members in international conferences, and to produce writtenmaterials on equity in mathematics education. Generous support from the Agencyenabled us to host the 1993 International Commission on Mathematics Instruc-tion's study of Gender and Mathematics Education.

Members of our network who work in schools and schools of education havehelped to disseminate information about research results on gender and mathe-matics as well as examples of successful intervention projects in the school system.Teachers have been provided a forum for discussing the issue of whether Genderand Mathenutics Education would be a suitable topic for examination in schoolsand teacher education.

What Are the Benefits of a Network?

Is it possible to achieve results by working together in a network? What are theoutcomes for women in mathematics? Several examples have already beenmentioned in the discussion above.

By creating contacts through the network, women have been able to use eachother in their work as speakers, reference's, sources of information and inspiration.As mentioned above, women's work is often made invisible, but through thenetwork members have been able to get to know about other women's work. For

63

Barbro Grei,hobn

example, during the past year the doctoral theses of three women in the networkhave been summarized in our newsletters. Together we work for greater equality%raised consciousness about gender and mathematics, the development of theschool mathematics curriculum Uohansson, 1993), and improved textbooks andother print and resource materials (Olsson, 1993).

Women must be permitted to contribute in mathematics and their perspectivemust be represented within the discipline and its organizing structures. There ispractically no literature in Sweden about women and mathematics. The con-ference proceedings from the Women and Mathematics conferences are now beingused as course literature by teacher education programmes at several universities.The network has put the issue of gender and mathematics education on theagenda. Through the network, women have been asked to participate andcontribute to conferences (Emanuelsson ci a/.. 1992, 1994) and working groups.as well as to the international exchange of ideas. Sweden has one of the mostgender-segregated labour markets in Europe (Sweden Statistics, 1994). One reasonfor this may be that preparation in mathematics (and scien,:e) are prerequisites forengaging in many professions. So to open doors for women to the labour market.we need first to open doors for them to mathematics.

The efkct of the Women and Mathematics network is much stronger than wecould have imagined when we began. It is like rings on the water after you havethrown in a stone. The results multiply and become important, not only forwomen, but also for men and for the discipline itself.

References

Aiu SK( R:( ;, M. and GR I yin )1 M. B. (1987) Matematikgranskuiv, Stockholm. Statens Institutfor Lironwdel.

)I I I , G. (Ed) (1994) keinnor och Matentatik. Kottferensrapport, Lulea, Lulea University.EMANIA I j( )1IANSON, B.. RI /s/ N, B. and Rvsnx. R. (Eds) (1992) Dokumentation

at. 7:e Matematikbietwalen, Goteborg, Goteborgs Universitet.EMA NU! I ss ix. G., .1( )1 IANS( B.. R( isi N, B. and RYI nx,. R. (Eds) (1994) Dokumentation

at' 8:e Matentankbiennalen, Giiteborg. Goteborgs Universitet.Fm xi MA. E. (1994) 'Mathematics. gender and research', in Gm vi nii M. B. and HANNA,

G., Gender and Mathematio Education, an IC.111 Study. Lund, Lund University Press,pp. 25-42.

(i& 151 R it to. B. (Ed) (1990) kvinnar och matetnatik. Kouleren.qapport. Malmñ, Lirarhogskolani Malmo.

Gni vi Kit Nt, B. (1991) 'Kvinnor och matematik-nAverk bildat'. Nanmaren. 18, 2,

pp. 43-4.Gni vi ior M. B. (Ed) (1992a) Kvinnot odt matentarik. Rapportet out rtbildniv, nr 1, 1992,

Lirarhiigskolan i Malmö.Git i vi ii M. B. (Ed) (1992h) 'Report about activities in Sweden. 1988-92', New,lettel of

the International Organization of libtnen in .l1athematio Education, 8, 1, pp. 14-15.vi nil M. 13. (1994) 'Svenska kvinnor i matematiken', in BRAtil n I G. (Ed) Keinnor och

latcmatik. konferettrapport, Lulea, Lulea University pp. 62-73.

64

A National IVetwork 414/omen

GRINI101 M. B. and Nit ssoN, M. (1994) 'Differential pet-fit-mance in mathematics at theend of compulsory schooling a European comparison: Sweden', in BUR l'ON, L.(Ed) Who Counts? Atsessing Mathematks in Europe, Stoke-on-Trent, Trentham Books,pp. 241-262.

HANNA, G. (1994) 'Should girls and boys be taught differently?', in BIHIL ER, R., SCISt RABI. IL, R. and WINK0 MANN, B. (Eds) Didactics of Matlwmatics as ,: Scientific

Dis4line, Dordrecht, Kluwer Academic, pp. 303-15.JolIANSSON, B. (1993) 'Utvecklingsgrupp bildad', Ninnnaren, 20, 2, p. 34.KIMBAI I , M. (1989) 'A new perspective on women's math achievement'. Psychological

Bulletin, 105, 2, pp. 198-214.LJUNG, G. (1990) Centrala prov i Matematik pa liukrna 1985-89, Stockholm,

Primgruppen.NAVE (1993) Sant,. ja. KotyiTenserapport, Oslo, Norges forskningsrad.

tisoti, G. (1993) 'Stöd och stimulansmaterial fir gymnasieliirare', Nanmaren, 20, 4,p. 11.

PIINp, R. (1994) 'Kvinnehge matematikere i Norden og i Europa idag och i morgon',in BRAN: )1 I I , G. (Ed) Kvinnor Matematik Koukrensrapport, Lulea, Lulea University,pp. 133-8.

RONNBACK, 1. (1992) 'Könsdifferentierad matematikundervisning i ak, 4-6', in Giu v-to, B. (Ed) Kvinnor och matematik. Rapporter on: Utbildning, nr 1, 1992, Malmo,

Cirarhogskolan i Malm6, pp. 24-37...Swi SiAl IS I ICS (1994) 1 tid och arid, Orebro, Statistike Centralbyran.TINGI Ft I, K. (Ed) (1991) Kvinder og matematik, Konkrencerapport, Copenhagen, Planlig-

ninpgruppen.NMARK, B. (1993) 'Visst det battre, men inte 1r det bra!' -irskrOnika 1992/93.

Lund, Lunds Universitet, pp. 4-5.

65

Chapter 7

Women Do Maths

Claudie Solar, Louise Lafortune and Helene Kay ler

Introduction

The title of this chapter derives from the title of the 1992 publication (Lafortuneand Kayler. 1992) of MOIFEM (Mouvement International pour les Femmes etl'Enseignement des Mathematiques), the French-Canadian section of IOWME.Since its inception in 1985. MOIFEM has dedicated its activities to promotingwomen in mathematics, developing feminist approaches to the learning andteaching of mathematics (in order to attract and retain more women inmathematics and related subjects), demystifying mathematics, and establishing aFrench-speaking 'women and mathematic.' network. Two conferences (Lafor-tune, 1986. 1989) were the first projects the group undertook. The conferences,designed to promote an understanding of women, mathematics, and mathematicseducation, generated three themes which recur in all our discussions andknowledge development: women and mathematics, the role of affect in mathe-matics learning, and the demystification of mathematics. The understanding wedeveloped through these conferences encouraged us to move from theory topractice in order to tackle the issue of teaching mathematics to women. Thischapter describes the development of this more practical aspect of our work.

'Maths Without Myths'

'Des maths sans les mythes' (that is, 'Maths Without Myths') is the title of theworkshop we designed for women involved both in the educational system and inwomen's groups. Our aims were to sensitize women to the myths surroundingtheir mathematical abilities, to promote pedagogical approaches which arcfavoarable to women, and to develop a network of women concerned about thestatus of women in mathenutics. Obviously, the process of developing theobjectives, cont...nt, approaches. and material, as well as the overall planning of*these workshops took much time and energy. Some material was adapted fromLafortune (1990), others were developed specifically for the workshops; ourfeminist perspective on pedagogy emerged from our various writings (Kayler and

66

Ri)/1101 Do Maths

Lafortune, 1991; Solar, 1992) and from our group debates. Before we beganworking with others, we had much work to do on ourselves it was an excitingtime for creativity and the sharing of knowledge.

The publicity across Quebec was very succe.s.fill and it was not long beforethe workshops were in high demand. We targeted people who either had directcontact with girls and women through the educational system, or were playing animportant role in shaping their academic or vocational orientations. Theparticipants, a few of whom were men, were drawn in roughly equal numbersfrom the following groups: teachers at all levels of the educational system;counsellors, psychologists, special educators and students; students enrolled inadult-education courses, and members of women's groups and communitylearning centres.

Each workshop lasted a full day and the structure was adapted to suit theabilities of the fiicilitators as well as the needs and characteristics of the participants.The day included five activities, chosen from a bank of twenty-three, and aimedat exploring the three themes we had developed in analysing the status of womenin mathematics. In the next section of this chapter we provide two examples fromeach of these areas. At the conclusion of each workshop, participants are providedwith descriptions of all activities and asked to reflect on possible follow-upworkshops they might conduct themselves within their own networks, schools, orcommunity groups. The purpose here is to promote further network develop-ment, as well as dissemination of the activities and approaches.

The evaluations of the workshops, both formal and informal, indicate thatthey have been very successful. Sonic participants suggested that all teachers shouldundertake such professional development, while others said that they would usewhat they had learned to improve their own teaching. Still others claimed to haverediscovered mathematics and mentioned that they might go back to studying thesubject. These commentsimong others, have provided us with encouragementfor continued work in the pursuit of equity for women in mathematics.

Sample Activities

11;nnen and Matimatics

Um. question d'image (Lafortune and Kayler, 1992, p. 108) is an activity whichbegins by asking participants to draw a scientist at work. Once the drawings arefinP;.d, they are displayed on the wall and each participant describes his or hercreation to the group. Following thisi discussion of popular stereotypes ofmathematicians and scientists ensues. Experience has shown that most drawingsdepict the scientist as maleIlthough the possibility of drawing a female scientistis explicitly suggested. People usually picture the scientist as a man with no socialor emotional life, no concern for poliOcs, working alone. He is often described ascold, objective, intellectually giftedind sloppy. Such an image is not likely toattract females to scientific careers. Holmes and Silverman (1992), in their research

67

Clandie Solar, Lmise Lqfortnne and HNne Kay ler

on Canadian teenagers, found that girls give significant consideration to theirphysical appearance and greatly value interpersonal relationships. Moreover, manyof the drawings illustrate the scientist as having a head but no body, thusemphasizing the centrality of the mind, the separation of mind and body, and theassociation of mathematics and science with male intelligence.

Um' histoire des mathematiciennes (Lafortune and Kayler, 1992, p. 51) is anactivity that challenges the view that there are no women in mathematics. Theintent here is to raise awareness of the historical absence of women, be it inmathematics or in any other field. First participants are asked to name scientists thatthey have heard about in the fields of mathematics, physics, chemistry or biology.After writing down the names, the small number, or total absence, of women onthe list is readily apparent. To facilitate these sessions, we developed an educationalposter which portrays women mathematicians of the past (Hypatia, Maria GaetanaAgnesi, Mary Fairfax-Sommerville, Sonya Kovaleskaya, Grace Chisholm Youngand Emmy Noether), as well as those of the future (illustrated by five girls doingmathematics at school). The function of this representation is to help focus thediscussion on the paucity of women in science, the reasons for their absence, andthe mummer in which society functions to exclude women from participating inmathematics. Information is then provided about the lives of several womenmathematicians, stressing the difficulties they encountered in pursuing theirchosen profession.

ilikctivity in Mathematics Learning

Renme-menimes (Lafortune and Kayler, 1992, p. 142) begins with a brainstormingsession centred on how participants feel about mathematics. The emotions arcrecorded and categorized as either negative or positive in nature, for instance,discomfort, fear, anxiety, frustration, and annoyance. or satisfaction. pleasure. joy,and comidence. Following this, the same strategy can be applied to specific topicsof mathematics such as trigonometry, computation, probability, geometry, calcu-lus, equations. graphs, functions or integrals. Alternatively, sheets of paper. eachidentifying a single topic of mathematics, can be displayed on the walls.Participants move around the room recording their feelings about each topic. Theficilitator then discusses the words used and encourages a general deliberation onthe role of affect and feelings in learning mathematics. This activity serves tovalidate the presence of emotion while learning mathematics, and reveals that thefeelings involved are not all negative, nor all positive, hut vary according to theindividual and the specific area of the discipline involved.

l'aime, pint(' pas (Lafortune and Kayler, 1992, p. 123) is another activitydealing with affect and mathematics. Numerous strategies can be employed. Oneof them entails having participants till out a questionnaire on their opinions andattitudes towards mathematics. The questions are evenly distributed among threeaspects: difficulty in learning mathematics, the value of mathematics, and thepleasure of doing mathematics. Once the participants have calculated their attitudescores, they are collected anonymously in order to provide a group portrait. In this

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litnen Do Maths

way it becomes possible to show that there are a variety of attitudes amongparticipants. The ensuing discussion focuses on what participants have discovered,the differences in their viewpoints, and more generally on the training they havereceived in mathematics. what they like or dislike about mathematics, how usefulthey perceive it to be, and what areas they find easy or difficult.

Demystification (fMathetuatics

An example of the kind of activity we use to demystify mathematics is .4utour du(Tule (Lafortune and Kayler, 1992. p. 102). the popular method of helping studentsdevelop an understanding of IC and how it relates to the circumference of the circle.(Participants measure the perimeter and diameter of the tops of various circularobjects and perform appropriate calculations to obtain an approximation of TOThis activity serves to communicate the meaning of TC in concrete terms.Thereafter, the participants can discuss the nature of mathematics as a sciencedeveloped by human beings, its use, how it is learned, and its history Through thisprocess. awareness of the significance of mathematics and the meaningless way inwhich it is often taught are highlighted.

Another activity in this area is Lc jogging matUmatique (Lafortune and Kayler.1992, p. 98). In contrast with those presented thus far, this particular activity isconducted over a period of time, for gradual build-up and regular repetition areessential. Its aim is to communicate the notion that learning mathematics does notrequire any particular gifts, rather the proper training and practice. The activity isconducted either at the beginning or at the end of each activity session.Participants are required to answer several types of questions using a definition, asymbol. an estimate, or the result of mental calculation. The questions beginsimply and gradually become more difficult. At first, the workshop leaderformulates the questions. but soon the women themselves can be encouraged tosuggest questions. In this way, women gain confidence in both the answering andthe problem-posing components of the activity. We encourage participants to keeptrack of their results so that they can observe improvement in their scores and thussee the benefits of practice. In conducting this activity it is important not toemphasize speed. Emphasizing speed can place participant, :ider great stress. aswell as increase their sense of failure and lower their self-e In. Speed is not thegoal of this activity. Time limits should be used cautiously only to illustrate onepossible desirable outcome of successt-ul training. After participating in this activity,one woman stated: 'I feel more intelligent. When people talk about numbers. Ipanic less and less, for even though I am slower than others. I succeed in havinga fur idea of what the result should be.'

.4 6 9

Claudic Solar, Louise Lativtune and Milne Kay ler

Conclusion

This project has alloWed us not only to develop, test and refine our workshopactivities, but also to produce other types of material. For example, we havedesigned four pedagogical posters dealing with various aspects of mathematics: oneon the 'ups arid downs', the emotions involved in learning mathematics; one onwomen mathematicians, past and present; one on being mathematically gifted; andone on mathematical formulas and symbols related to real-life situations. Inaddition to these four creative endeavours, designed for use in training sessions, weproduced another poster to promote women in mathematics as well as MOIFEM.It displays forty-nine photographs of women demonstrating that they participatein mathematics and evokes the image that there are infinitely many women 'doing'mathematics.

We conclude by giving a brief description of our current and proposedprojects. Following the workshops, we studied the manner in which women 'do'mathematics in their lives, thus approaching a feminist critique of the discipline.We have examined how mathematics is involved in traditionally female activitiessuch as lace-making, macrame, and quilting (Barrette and Lafortune, 1994). Wehave further refined our description of feminist pedagogy by presenting teacherswith case studies of girls in high school and analysing their reactions (Solar, 1994).We have also re-examined the research on sex-related differences in the area ofspatial visualization (Pallascio, 1994) and explored the way Inuit women usemathematics and teach it to their offspting (Paquin and Oovaut Putayuk, 1994).We have studied how mathematics is used in qualitative and quantitative research,including feminist research (Lafortune, 1994). Finally, we have used the testimonyof women mathematicians to discuss their experiences with mathematicalprocesses (Kayler and Caron, 1994). (For a detailed account of all these pursuits seeSolar, 1994 and Lafortune, 1994).

References

BARRI.1 I I., M. and LAI OR I LIN! , L. (1994), 'La dentelle mathematique'. in SOI Alk, C. andLAI oit UNI., L. (Eds) Des Mathematiqiws Autrement, Montreal, Les editions du remue-menage, pp. 115-69.

HOI MI S, J. and Su vi RMAN, E.L. (1992) J'ai des Choses a Dire ... tamtez--moi!, Ottawa.Conseil consultatif eanadien de la situation de 1,1 femme.

KAY! I It, H. and CAR( R. (1994) 'Les femme% et l'activite mathetnatique', in SOI AR, C.and LAI ok UN! , L. (Eds) Des AlatIOnatiques Autronent, Montreal, Les editions durentue-menage.

KAY!! IL, H. and LAI clt I UNI., L. (1991) 'Pedagogic feministe en mathenutiques', iiducationdes Femmes/ If imien 's Education, 9,2, pp. 27-30.

LAI (IR I t , L. (Ed) (1986) Femmes et A lathi.matiqur, Montreal, Les editions di: reinue-menage.

1 AI 1NI , L. (Ed) (1989) Quelles Difli'rences? Les Femmes et l'Enscigmment des Mathima-fives, Montreal, Les editions du mime-menage.

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144wien Do Maths

Livo im UNA., I.. (1990) Demythification de la Mathematique. Materiel Didaaique. 'Operationbottle I mythes', Québec, Ministere de l'enseignement supérieur et de la science.Direction generale de l'enseignement collegial.

LAI olu LINE, L. (1994) 'Femmes, recherche et mathematiques'. in SOI Alt, C. and',AMR EUNE, L. (Eels) Des Mathetnatiques Autrewnt. Montreal, Les editions du remue-menage. pp. 67-113.

LA1-010 UNE, L. and KAY1 Eft, H. in collaboration with BAIUU I. I F, M.. CARON, R.,PAQUIN, L. and Soi Alt, C. (1992) Les Femmes Font des Maths, Montreal, Les editionsdu remuc-menage.

PAI ['WIC/ R. (1994) 'Visualisation spatiale et differences(?) scion k's sexes', in Sol AR, C.and LAI-01011Ni, L. (Eds) Des Mathematiques Autrement, Montreal, Les editions du!Time-menage, pp. 171-271.

PAQUIN, L. and 00yAli I PI: AYUR, P. (1994) 'Inuttitut, mathematiques et Einmes% inSol AR, C. and LAI OM l/NI , L. (Eds) Des Mathematiques Autrement, Montreal, Lcseditions du remue-menage. pp. 275-316.

Sot AR. C. (1992) 'Dentelle de pedagogics féministes'. Revue canadienne d'edwation/Canadian found( q' Education, 17,3, pp. 264-85.

SOI Alt, C. (1994) 'Femmes. mathematiques et pedagogic'. in Sot AR. C. and LA! Olt I UNE.L. (Eds) Des Mathematiques Autrement, Montreal. Les t:ditions du remue-menage.pp. 23-43.

SOI AR, C. and LAI OKI ONE., L. (Eds) (1994) Des Mathematiques Autrement, Montreal, Leseditions du renme-menage.

71

Chapter 8

'Girls and Computers': MakingTeachers Aware of the Problems

Cornelia Niederdrenk-FeIgner

Introduction

'Girls and Computers' is the title of a project. undertaken by the German Institutefor Distance Education, to develop study materials for the in-service training ofteachers who wish to use computers in their teaching in areas such as mathematics,writingind social studies. At the same time, the muter ial will be of use to teachersof a course, 'Basic Education in Information Technology', now commonly taughtin all parts of Germany.'

Our starting point is the well-known fict that there are gender differences inhow young people use and approach computers. In recent years, various studiesand action research on gender issues in the use of computers have been publishedin Germany (see for example, Faulstich-Wieland and Dick, 1989; Heppner,Osterhoff and Schiersmann, 1990). The 'Girls and Computers' project is based onthese findings with the aim of changing the classroom experience, primarily bychanging teacher behaviour, in order to give more emphasis to the competenciesand interests of girls.

The place of-computers in the classroom is a topic of much discussion amongmathematics educators. However, gender issues has been neglected in the Germandiscussion (see. for example, Kaiser-Messmer. 1989). When computers are used inteaching mathematics or other school subjects, gender differences become veryobviousind our hope is that by discussing gender issues in relation to girls andcomputers, We Will call attention as Well to the causes of gender difkrences inmathematics.

Causes of Gender Differences

In the context we considered relevant for our project. the following appear to bethe most important reasons for gender differences. First, there are markeddifferences in the ways boys and girls are brought up in Germany and,

'Girls and Computers'

consequently, also in their adaptation to societal expectations and requirements asmanifested through the education system. The development of young people isinfluenced in a variety of ways by sex-role stereotypes. Research with schoolchildren (Horstkemper. 1987) has shown that girls tend to be significantly lessaware than boys of the potential of their abilities and skills, even though theirschool performance is significantly superior to that of boys. Schoolgirls who dosvell tend to think that they have 'just been lucky', rather than ascribing it to theirown ability. When confronted with a school environment that is dominated bymales and masculine values, girls tend to underestimate and give second place totheir own abilities and skills, rather than demonstrate them with confidence. Malesdefine the knowledge that is worth being taught, 'masculine' interests rate higherthan 'feminine' ones, and boys receive more attention in the classroom than girls(Enders-Drag:isser and Fuchs, 1989). As far as the teaching of mathematics isconcerned, Srocke (1989) has demonstrated very clearly the extent to which thecontent and presentation of this subject have been modelled on 'masculine'orientations.

To make matters worse, teenage girls find themselves confronted withcontradictory expectations. They are expected to plan their lives so as to combinethe requirements of whatever occupation, vocation or profession they choose,with those of 'feminmity' (Horstkemper, 1990). but the qualities that are requiredfor success in a career (that is, the ability to achieve one's own objectives, 'rational'approaches) are not valued highly and are not accepted easily from v omen. Inmany occupations and professions, particularly those from which women havebeen traditionally excluded, women are now confronted with standards and normsthat have been defined exclusively by men. All these factors provide an additionaldisadvantage to girls and young women in defining their own 'female' or'feminine' identity (Hurrelmann, 1991).

Second, established sex-role stereotypes in German society betray a strongconnection between technology and 'masculinity' on the one hand, andincompatibility between technology and 'femininity' on the other. This meansthat, since women are considered incapable of coping with technology andtechnal appliances, when it is evident that the opposite is true, such women arelabelled 'not really feminine'. The relationship between women and technologyis always defined in terms of failure to contbrm to a standard that has been set bymen. A lack of skill or interest in technical or technological matters is regarded asa 'feminine deficit', competence and interest in these matters as a non-femininedeviance from an expected sex-role stereotype, whereas a more neutral view istaken in both cases for males. A widespread belief that men are the 'developers'and women the 'users' of technology prevails (Faulstich-Wieland, 1990). Andthis 'divisit:n of labour' adds an additional dimension to the problems alreadyoutlined.

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Cornelia Niederdrenk-Fekner

The Problem

The 'Girls and Computers' project is very different from other German in-serviceteacher-training projects, both those designed to address specific school subjects,such as mathematics, and those designed to address issues, for example, pupil--teacher interaction. The problem we face in dealing with the issue of girls andcomputers is that most teachers are not even aware there is a problem.

Sex-role stereotypes are not confined only to students, but are also evident inthe minds of teachers of both sexes and influence their behaviour. Theirinteractions with young people and their different expectations of boys and girlsare strongly influenced by ideas on what males and females 'should' be doing.Young people, in turn, are influenced by their teachers' behaviour. They 'learn'to behave in conformity with gendered expectations. This vicious cycle can onlybe effectively broken by changing teacher behaviour.

There is no point, however, in simply demanding a change. Teachers have tobe convinced, first of all, of the need for change. In addition they should beprovided with opportunities to check, understand and modify their behaviour.This is a difficult matter to address because it is also a highly emotional one.Teacher groups that are pr,-clominantly male, as in the case of mathematicsteachers, tend to avoid, or even be 'hostile to, confronting this question. Even thefindings of serious academic research are discredited as irrelevant, with the claimthat an 'abstract' subject like mathematics leaves no room for discrimination againstfemales. On the other hand, making the observation that the same patterns ofgender-specific behaviour occur between male and female teachers in teacher-training seminars, has led to interesting discussions with participants talking aboutissues that concern them personally.

The `Girls and Computers' Study Material

Our teacher-training materials are designed for use in a variety of subject areasincluding mathematics, computer studies, German and social studies. Theyinclude a 'Basic Information' unit (Niederdrenk-Felgner, 1993) including back-ground information on factors influencing the development of students' skills andabilities during adolescenceInd other units, to be published in the next coupleof years, which exemplify 'girl-friendly' lesson plans and ideas for classroomact'xities which involve computers, some of them accompanied by software.

The 'Basic Information' unit is based, wherever possible, on concreteex,--nples. The idea is to sharpen teachers' awareness, and to highlight concretesituations where teachers and pupils display 'gender-specitic' behaviour. Bearingin mind that teachers have their own personal beliefs and explanations, their ONVI1'subjective theories' about problems, each Lhapter begins with a case study a

description ot' a real classroom situation, or a relevant quotation followed byquestions. In this way, teachers He encouraged to explore various aspects of thecase study and to formulate appropriate strategies for taking action. A variety ot'

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'Girls and Computers'

questions are included to assist teachers in exploring the case studies and latersections in the text refer back to sonic of these questions, answering them wherepossible. A different type of question encourages readers to reflect on their ownclassroom situation and make their own observations. This may include assign-ments, like determining the number of male and female mathematics teachers atone's own school, or analysing school textbooks for examples of cliches and sex-role stereotypes. The idea is to enable readers, wherever possible, to make theirown assessment of problems and to verify facts that have been cited in the text.Rather than being mere consumers of the text, readers are encouraged to checkand seek out opportunities for change in their own schools.

Another important aspect of our methodology is to encourage teachers to takethe personal experiences of their own pupils into account. The material containssample essays for analysis. For example, essays on the topic 'A normal working daywhen I am 30 years old', written in test classes, show very clearly the differencebetween boys' and girls' perspectives in planning their futures, and the extent towhich their ideas are based on cliches and stereotypes. Another essay, entitled'What I have done with a computer. Why I enjoyed it/did not enjoy it', givesteachers an idea of the extent to which pupils in a beginners' class have computerexperience. At the same time, these essays reveal the eypectations, attitudes, Ind,in some cases, fears of girls and boys. Teachers ap .ncouraged to ask their ownpurAls to write similar essays and to start similar discussions in their own classes.

In the section on classroom activities, we point out why merely reofganizingclasses, for example by separating boys and girls, is no reipedy in itself. However,we also point out how, at certain stages and when combined with discussion of thepros and cons of coeducation, temporary instruction in gender-separate groupsmay be a productive stimulus for non-stereotypical classroom behaviour.

What We Are Hoping to Achieve

The ideas on which this material is bawd have already been tried out in seminarsfor practising teachers. These were attended by teachers who had specificallyapplied to participate in a 'Girls and Computers' seminar although many of themwere initially quite sceptical, believing, for example, that there was nothingunusual in the way males and females are presented in school textbooks, despitethe obvious evidence to the contrary (men/boys appear much more frequently andin more interesting contexts than women/girls). It was only after sonic discussionthat these participants were prepared to admit the existence of difference,. and theimplications.

As we had expected, all participants claimed that there was no difThrence inthe way they treated boys ;lad girls in their owl classroom, or at least that they didnot intend to make any significant distinction in their student interactions.Presenting the teachers with the findings of research projects was n It sufficientlyconvincing. What was much more effective was involving them in a classroomobservation and follow-up analysis of an actual lesson that took place during the

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Cornelia Niederdrenk-FeIgner

seminar. By the end of the seminar, all participants understood why the issue wasimportant, and were prepared to reconsider their cwn classroom behaviour.

Our material is also intended for use in the context of vocational training ofyoung women in technical occupations. Our ultimate hope is that this type ofapproach will contribute to long-term changes and improvements of attitudes.

Note

1 I wish to thank Lothar Letsche and Helga Krahn, the other members of the projectteam, for their helpful and constructive comments and discussions. Lothar Letscheprovided the English-language version of this chapter.

References

ENI,I.Its-DRM;ASSIA, U. and Fuci is, C. (1989) lnteraktionen der Gesthlechter: Sex-

ismusstrukturen in der Schule. Weinheini und Munchen, Juventa Verlag.FAUI S I AM), H. and l): K. A. (1989) Madcluwbildung und neue lithttologien:

AbschlulThericht de, wissenschafflichen Begleitung z-um hessischen Vorhaben, Wiesbaden,Hessisches Institut far Bildungsplanung und Schulentwicklung (HIBS), SonderreiheHeft 29.

FAUI Si ICI ANI ), H. (1990) 'Technikdistanz von Madchen?', in HORS I , M.and WM:NM-Wits:11,M IAG1.11., L. (Eds) Madrhen und Jungen Manner nod Frauen in

der Srhule. Die Deutsche Schule, I. Heihcft, pp. 110-25.HILPPNER., G.. Os i i.iu lot i; J. and SCIIIIASMANN, C. (1990) Computer? '1nteressieren tat's

schon, alter . . Bielefeld, Kleine Verlag.F14.)ksi KFMITIL, M. (1987) Schule, Geschlecht und Selbstvertrauen: Eine Lawschnntstudie fiber

ladchensoz-ialisation in der Sthule. Weinheim und Miinchen, Juventa Verlag.

HoRs I K1 It, M. (1990) 'Zwischen Anspruch und Selbsthescheidung Berufs- und

Lebensentwiirfe von Schiderinnen,' in HORS 1 KI.MIT.11, M. and WAGNER-WIN I MI IAGElt, L. (Eds) Madclwn und Jungen Manner und Frinten in Set Sthule. Die

Deutsche Schuh., I. Beikfi, pp. 17-31.MANN, K. (1991) lunge Frauen: Sensibler und selbstkritischer tic junge MAnner',

Padagogik, 7-8, pp. 58-62.Kmsi ssmi It, G. (1989) 'Frau und Mathematik chi verdrangtes Thema der

Mathematikdidaktik', Zentrardatt.für Didaktik der Lithemarik, 21, pp. 56--66.

Nita )1 ILI )14.1.NK-F1 I ( , C. (1993) Compwer im koedukativen t'nterricht: '..;tudienbrief MIS der

Reihe 'Madclu'n um! Computer', Tiibingen, German Institute for Distance Education atthe University of Ti'ibingen.

tim , B. (1989) Madcluw und Mathematik. Wiesbaden. Deu6cher Universitats Verlag

76

Chapter 9

Common Threads: Perceptions ofMathematics Education and theTraditional Work of Women

Mary Harris

Introduction

'Common Threads' was the name of a unique exhibition that began as a small off-shoot of a mathematics education project. The result of a pa;sing remark by acommittee menther, the exhibition was researched, designed, and built by theproject director, almost single-handedly, in her spare time. 'Common Threads' wasabout mathematics, that cerebral and stereotypically masculine discipline, yet all itsexhibits were needlework, that practical and stereotypically feminine pursuit. Bypresent:1g these two extremes in terms of each other, 'Common Threads'demon_zi.ated the wealth of mathematical thinking and skill that wes on withinthe very practices that traditionally mark their ..atithesis.

The title 'Common Threads' was a deliberate play on words chosen toillustrate the shades of meaning in the exhibition's message. Thus, the word'common' was used in the sense of ordinary, everyday. "as well as in the sense of beingwidespread. The word 'threads' referred to the yarns of knitting, the silk ofembroidery the grasses of a basket, and the warp and wcft of weaving. But it alsoimplied the developmental strands of mathematics, both the vertical paths throughwhich an area of mathematics develops in depth, and the horizontal links betweenvarious branches of the discipline. In keeping with the cloth analogy, 'CommonThreads' represented a seamless robe of mathematical activity within a v,!ryordinary part of anyone's daily life, anywhere on earth.

exhibition was originally displayed for just one week, however, its impactduring that week was such that it subsequently toured England for a period of twoyears. In response to international intereq, two copies of a new version were latercreated by the British Council, and these travelled to more than twenty countriesin Europe, Africa, Australasia and America over a period of four years. During itstours, 'Common Threads' has stimulated curriculum development in schoolmathenutics, cross-curricular work from elementary to tertiary levels, vocationalwork in formal and informal settings, ethno-mathematics research, and pro-grammes in women's development.

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Mary Harris

Two Exhibitions

The new version of 'Common Threads' was substantially different from itspredecessor, although the original title was retained. For the sake of clarity in thischapter, the original will be referred to as 'Common Threads l', and theredesigned versions will be termed 'Common Threads 2'.

'Common Threads 1' was designed and created by the project 'Maths inWork' which I directed from 1981 until its completion in 1991. It w. anetilicator's homemade exhibition, the outcome of a particular perspective inmathematics education discussed in more detail below. 'Common Thrcads 1'consisted of five sections: symmetry, number, creativity, information nandling, andproblem-solving. The symmetry section, which set the tone for mathematicalcontinuity in breadth and depth throughout the entire exhibition, was developedmathematically through panels explaining the ways in which mathematicians thinkof space, including references to symmetry and transformations. It began with apiece of cross-stitch embroidery that I had purchased in a craft market while onholiday in the former Yugoslavia. I illustrated the processes whereby anembroiderer develops mathematical symmetry on cloth used like graph paper, bymaking a set of small embroidered panels stopped at different stages of the designdevelopment. My comments as to how I, the embroiderer, had built up thepatterns formed the captions, and I placed mirrors beside the exhibit so thatvisitors couid explore the design stages for themselves. I illustrated strip patternsusing hair ribbons and pieces of upholstery trinmiing, and the section ended witha mathematical description of the wall-paper patterns shown on dress materials anda Turkish kilim rug. The concept of symmetry was not confined to this section,however, but overlapped with the other areas to demonstrate its significancethroughout mathematics. For example, the number section, primarily illustratedthrough knitted objects, contained Aran and Fairisle sweaters whose symmetricaldesigns were analysed in terms of number groups.

'Common Threads 1 broke new ground as a learning resource, but, as I waslater to discover, it also broke at least two rules of exhibition design. First, sincecloth is tactile and inviting, it never occurred to me to say 'Do Not Touch% onthe contrary, I intended the materials to be touched. I also provided a table withgraph paper, pencils, and other materials people find helpful when doingmathematics. In the event, nothing was damaged or stolen, indeed as theexhibition toured, people sometimes even added to it. Evidentl),, the exhibit wastreated with affection as well as with respect, and it sometimes returned to basewith artifacts carefidly mended.

Second, I did not know the received wisdom on captions in the field ofprofessional exhibition design. Since the purpose of my captions was to offermathematical explanation of what transpired in the making of the cloth items, Iwrote them of a length necessary to do just that. In order to make the captionscomfot table to read, I did place text and diagrams in inviting ways, but it neveroccurred 10 me to limit their length. In the case ofthe syninietry section, for example,sonic of :he captions extended to whole panels. However, their effectiveness was

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Common Threads

demonstrated by the classes who visited and revisited 'Common Threads 1' as part oftheir school work on transformation geometry. Members ofthe public also carefullyreferred to them, taking time to read, analyse, and understand.

The two 'Common Threads 2' exhibitions were smaller, but dramaticallymore beautiful, and built specifically for overseas touring by professional designersfrom the British Council Exhibitions Department. An immediately obviousdifference between 'Common Threads 2' and the original exhibition, apart fromthe professionalism of the design. was that all the tactile materials were now behindperspex. More subtle, however, was that the captions of 'Common Threads 2'demonstrated a perception of mathematics education that differed from theoriginal. Not only was the mathematics content considerably reduced on aestheticgrounds ('Common Threads 1 captions were far too long!), but the designers feltthat the general public would not be able to cope with the depth of mathematicsexplanations originally offered. The combination of the reduced captions and thespectacular cloth the new designers produced effectively fragmented the exhibit.Thusilthough there was still some overlap between the new sections of symmetry,geometry, number, and codes, over 411 there was much less mathematics and it waspresented at a more superficial am.f disjointed level. Whereas 'Common Threadsl' had been an interweaving of mathematics and cloth, 'Common Threads 2' wascloth with mathematical comment. The view of mathematics as low level.disconnected, and mainly arithmetical techniques applied to various situations, isalready the predominant public perception and 'Common Threads 2' was madeto tit it. In addition, design considerations now took priority over educationalon, -, and the exhibition was no longer mine. The professional understanding ofmathematics as a process which thc exhibition was originally intended to conveyhad slipped away, aud I was forced to devise support materials to enable 'CommonThreads 2' to be used in the same way that I had used the original exhibition.

Work as a Resource

The project that generated 'Common Threads 1 'Maths in Work', w is designedto investigate the skills new school leavers employed in their first jobs. By meansof a questionnaire survey, the idea was to use the data generated to monitor theeffectiveness of a new secondary school-leavers course, as well as a source ofrealistic learning materials for schools. A problem I struggled with, as projectdirector, was how to reconcile behavioural data generated by .1 vocationallyoriented piece of research with my constructivist philosophy of whool-mathematics education. I have discussed this issue in detail elsewhere (Narris,1991). Suffice it to say that, the experience of analysing the data. and of findingmore mathematics in the answers to questions that were not about mathematics thanin answers to questions that were, eventually led me to abandon the research dataas a true picture of mathematics in the workplace. I replaced it with my ownparticipatory observations which, in turn, led to a complete rethinking of thelearning materials I was to produce.

6 ,J 7 9

Mary Harris

A feature of tly: research questionnaires was that the questions genmited twosorts of response. a numerical frequency score of how yien a particular skill was used

and a description of the contexts in which frequently used skills were used. Theitem about context was included as a check that the worker had understood thequestion. By analysing this context data in addition to the skills frequency results,

it was clear that workers were using many mathematics skills in their workalthough they might not admit to it when asked directly. For example. thehairdressers in the sample, when asked how they used ratio and proportion in theirwork, denied using them. However, when asked about how often they mixed

substances, they explained that it is essential to mix hair dyes in the Jorrectproportions otherwise a customer's hair could be damaged.

Two aspects of this and of many other similar responses were significant. First,

there seemed to be a general negative attitude towards school mathematics. Many

of the young workers had not enjoyed their school experience of mathematics andtended to respond negatively to the questionnaire as though they viewed it as a test.Second, they often did not connect what they did at work with what they haddone at school, nor did they expect to. The demonstrable understanding ofproportion in a practical situation displayed by the hairdressers raised noconnection in their minds with the algorithms of school mathematics. Manyexamples of this phenomenon led me to think that mathematics at work onlybecomes mathematics when workers think they cannot do it.

It was this distorting emphasis on skills as a picture of mathematics in theworkplace that caused me to abandon analysing the skills data and instead to take

to the workplace myself. Since negative attitudes to mathematics are so wide-spread. I tried not to mention the word 'mathematics' at all when asking my fellow

workers about their workplace problems. People love talking about their problemsand it was very interesting to note that the resolution nearly always contained somemathematical thinking, though often not of a sort included on the skills list ofresearchers in the area of mathematics and work.

The assumption that mathematics in the workplace at this level is whollyarithmetical is widespread. it appears, for instance, in the Cockcroft Report (1982)

in the chapter on workplace mathematics, where almost every use of the word'mathematics' actually means 'arithmetic'. Of course, underlying assumptionsinvariably condition the answers to questions. Actually doing the work, rather thanasking questions about it from a preconceived standpoint, reveals a much broader

picture.

If you are a scaffolding rigger, you don't do any calculating while you arerigging (and wouldn't Cockcroft say therefore that riggers don't usemuch maths?). But if you haven't grown the calibrated eyeball that knowsabout stresses and load bearing and bracing and moments and estimatinglengths and interpreting plans you won't last long as a rigger, nor will theperson below you. (Harris, 1985, p. 44)

The same applies to cardboard box makers:

Commoit Threads

The processes that a designer of cardboard boxes, or a designer of anythingelse, works through are those that a mathematical problem solver goesthrough. They start by researching the topic, exploring the parameters,then doodling ... before going on to choosing what seems like a goodway, exploring it. refining it, rejecting it if it doesn't work and eventuallycoming up with something satisfactory. I would argue that these processesgeneralise to other problematical situations whereas the specific skillsmaybe don't just because they are specific ... (Harris, 1985, p. 44)

The key is to ask questions not about mathematics skills on the job, but aboutworkplace problems, for 'by asking questions about where the gremlins usuallystrike, you get not only a true picture of what life is like but many examples of theproblematical situations which demand the real thought rather than the routinecalculations which don't' (Harris, 1985, p. 45).

As my experiences as a worker grew. I came to view job sites as 'resources' formathematical thinking instead of 'destinations' for the.application of mathematicslearned in school, a complete reversal of the accepted practice. The implicationsof this view for the design of teaching materials were also uncompromising. At thattime, learning materials for mathematics at the school-work interface were of aparticular kind. Typically, they took a skill, often something that had caused pupilsand teachers considerable pain over a number of years, such as the addition offractions, and then demonstrated the use of those skilh at work. The assumptionwas that by placing the skill in context, the pupil would see the necessity oflear....e; it, as well as come to understand its use. The educational reality, however,was the presentation of a distorted, limited and limiting view of mathematics toa group of already disenchanted pupils.

The authors of such materials inay or may not derive their inspirations fromreal workplaces. If they did, they tended to edit out variabks so as ) make themathematical skill component of the question more salient. If they did not, theyinvented hypothetical workplace situations, again for the purpose of illustrating themathematics. In either case, the materials were not a true representation of reality,and the first people to recognize this were the pupils. Since I had reversed this wayof doing things, I had to find a w ay of working with the entire context of theworkplace as a genorator of problems, not as a setting for the practice of isolatedskills taught separately and then applied like a bandage to a wound.

Perceptions of Mathematics Education

The first pack oflearning materials produced by the 'Maths in Work' project beganon the floors of cardboard-box fictories and generated a set of 'design and make'problems for mathematics teachers. It was called 'Wrap it Up' and contained abouttwenty-five problems in a form that could not possibly be mistaken for amathematics textbook. Each open-ended problem was presented with a minimumof instructions on a single sheet of paper. The format was such that teachers could

Mary larris

select problems, and adapt them to the needs of their own students. In this way,the role of the teacher became one of professional educator as opposed to skillstrainer. Because the problems were real, the materials did not talk down toanybody and there was no suggestion in either form or content that the work waslow-level; indeed, the level was what the student wanted it to be. The pack provedso successful with teachers that I developed the model for other workplaces.

The origin of the type of research and learning materials that I came to rejectlies in the social history of the British mathematics curriculum. I refer, here, to asociety so obsessed with social class, that it produced entirely different sorts ofmathematics education for different pupils based on their societal standing (see, forexample. Williams, 1961, for a very readable account of this period). The legacyof liberal mathematics (which of course contained Euclid) for the upper classes ofsociety, and utilitarian mathematics for the lower classes, can still be witnessedtoday in the disconnected skills included in the British National Curriculum andthe debate that surrounded them (National Commission on Education, 1993). Itis evident too in the manner in which companies recruit different groups ofworkers. At all levels, mathematics knowledge is still regarded as both a criterionof intelligence and a mark of the type of education a candidate has received. Acompany will recruit graduate mathematicians, not because it needs theirmathematics, but because a mathematics degree is evidence of the 'best sort ofbrain'. At the bottom end, 'unskilled' jobs and the 'basic skills' of school arithmeticare virtually definedin terms of each other. In fact, the closeness ofthe relationshipresults in its being unquestioned and therefore reinforced by much of the researchaimed at investigating it.

'Common Threads l' and the English 1)1ff

Different perceptions of mathematics education have had different effects atdifferent times. A good review of five educational ideologies and their philosoph-ical foundations and implications is provided in Ernest (1991). In the 1970s.increased employer concern about the low 'attainment of school leavers stimulatedresearch in this area. By the mid-1980s, two primary concerns in mathematicseducation were centred on gender and culture. The research question at that timewas 'Why are so many girls and members of ethnic minorities outperformed inschool mathematics by dominant culwr,: boys? In seeking to address this questionthrough the materials I was developing for my second learning pack, I looked forsomething that was as ordinary as the cardboard of the first pack. but wasparticularly resonant with girls of any culture.

Cloth is such a substance. It is so ordinary that it is taken for granted by those notiii the textiles business. Everywhere in the world people wear clothes, and these areusually produced by women at home or in a factory. In fact, the productive work ofwomen is of greater importance in the textiles industry than in any other trade(Holland, 1991). Work with textiles in all locations, and in most cultures, isthoroughly gendered. Thus, if I could demonstrate that the very nwdium whichdefines the lowly work of women worldwide is richly mathematical, then I could

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Common Threads

explore a fresh perspective on gender, culture, and mathematics education.Work began on a resource pack using the same format as 'Wrap it Up'. All

the problems were taken from the textiles workplace, and as the work progressed,the 'Maths in Work' office began to fill with interesting cloth from all over theworld. It was this collection that inspired the chance remark that led to the'Common Threads' exhibition. Research for the exhibition was an extension ofthe work already under way. As an experienced amateur needleworker myself, Ialready had practical experience regarding the mathematical basis of knitting,sewing and, weaving. Because I was also experienced in mathematics, it was arelatively simple matter to rework particular pieces of needlework using thetechniques and materials of the original worker, while simultaneously bringingout the mathematical thought inherent in the design development. For example,adapting a knitcing pattern to produce a garment of an intended size and shape isentirely a matter of ratio; designing a decoration for a garment is a question ofgeometry and number manipulation; matrices form the mathematical foundationrequired for weaving.

My lack of experience in the industrial sphere made research in this area bothchallenging and rewarding. It was here, however, that I was immediatelyconfronted with the public perception of mathematics in 'unskilled' work as beingnothing more than arithmetic. At almost every factory I visited, I was greeted withtirades against schools for not teaching long division, or by complaints that schoolleavers cannot multiply fractions. A visit to a tie factory was but one example andwill serve to illustrate. On arrival at the factory' and after the usual opening rounds.I was invited to look at some new software in the accounts department, thepresumed destination of my visit. Only after a considerable amount of persuasionwas I allowed to visit the cutting room. Here. I found workers manipulating thecloth from which ties were to be made, 50 as tO cut the three pattern pieces.without wasting any cloth, in such a way that the motifs on the finislrxi articleswould be horizontal and in the correct place. In direct contrast to schoolmathematics in which geometry is done on sheets of paper in portrait orientation(with triangles sitting firmly on their bases). the whole business of making ties isgeometry for real at the odd angles of real life, demanding sophisticated mental andphysical maMpulation of spatial images and artifacts. Readers are invited to unpicka necktie and investigate this for themselves.

At the opening of 'Common Threads', tl. enthusiasm was so great that it hadto be kept in check for safety reasons. Schools had to be turned away because thegaller y'. which was also open to the public, was not big enough, and it was for thisreason that the exhibition went on tour. Enthusiasm from members of the publicwas also marked. On one occasioni child of about 4 years of age stood at the door,grabbing the hands of adults as they entered and insisting that they 'come and lookat these lovely patterns'. An Indian academk, walking past the gallery on her wayto a meeting, wa, drawn to the exhibition by the mathematical analysis of thedesigns on the hem of her sari.

Once on tour, enthusiasm continued. Schools, craft workers, and passingvisitors worked in concert to add local colour, and a number of cross-curricular

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Mary Harris

projects were taken up within the schools. In one location, some lace makerscomplained that there was not enough lace in the exhibition and, in response,brought their own work to the exhibition and performed live demonstrations. Atthe end of one working day, they left their equipment on the table despitewarnings that children would touch. Their response was 'Of course. How else willthey learn?'

As 'Common Threads 1' toured, it became clear that it did respond to theissues it was designed to address. It raised the confidence of female pupils andteachers by confirming the nature and ext-mt of the mathematics in 'their' work.It confirmed the traditional narrowness of school teaching and textbooks, andexposed the gendered nature of the examples they offer. No longer was it possibleto deny the richness of mathematics that exists within the cultures of children whowere not achieving highly in school mathematics. The question 'Why are so manygirls and members of ethnic minorities outperformed in school mathematics bydominant culture boys?' changed to 'Why is school mathematics still so narrow-minded?'

The results of the tour and this change in questioning were reported in 1988at ICME-6 in Budapest where they attracted the attention of delegates from anumber of countries, notably Australia and New Zealand. It was requests, byeducators from these countries, to borrow 'Common Threads' that led the BritishCouncil to take the imaginative leap of redesigning 'Common Threads' and takingit on tour.

'Contmon 'Threads 2' and the Overseas Tour

The two copies of 'Common Threads 2' began touring at the end cl 1990, onein Denmark. and the other in Turkey. After Denmark, the first copy went on toNorway and then south to Nigeria. After Cameroon, Uganda, Malawi. Kenya,Zimbabwe, Botswana, Lesotho, and Swaziland, its journey concluded in Tanzania.The second copy headed south-east from Turkey, to Thailand, Malaysia,Singapore, Australiamd New Zealand, after which it crossed the Pacific to Braziland finally Canada.

Limitations of space do not allow for a detailed elaboration of the variety ofwork associated with the 'Common Threads 2' tours. The ways in which it hasbeen used have depended on both the educational policies and the perceptions ofits hosts. Gender and mathematics issues are a matter of culture and vary fromcountry to country, including those in which their existence is officially denied.When 'Common Threads 2' was provided to the mathematics educationcommunity of a particular country, it acted either as a resource for work alreadyin progress, or as a stimulus for new investigations. For example, in Botswana, theuniversity collaborated with the British Council in developing a series ofprogrammes for schools, and eventually a book of cross-curricular activities basedon Botswanan culture (British Council, 1993) which was distributed to everysecondary school in the country. In Thailandin exhibition of Thai textiles,complete with their own captions, was designed and set up alongside 'Common

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Common Threads

Threads 2' in two locations, a department store and an annual conference ofmathematics and science teachers. This heightened awareness of the localenvironment as a resource for learning, and has resulted in a textiles andmathematics display at every annual teachers' conference in Thailand since thetime of the original exhibition.

Discussion

The very of academic work divides and isolates studies that ought to beallowed to cross t-ertilize. The effectively separate histories of women's education,textiles work, and mathematics education make interesting parallel reading (Alic,1986; Boyer, 1989; Parker, 1984). Parker, in her book The Subversive Stitch (1984),explores and interprets the processes whereby embroidery. the highly respectedmedieval art of both men and women, was down-graded to a craft, hierarchicallygendered through power-brokering and social control, to the point whereembroidery became almost a genetic character;stic of nineteenth century feminin-ity. The first use of embroidery in the curriculum was to differentiate girls'education from that of boys', a means whereby girls could be educated withoutposing a threat to boys. By the late nineteenth century, its place in the curriculumhad become vocational needlework for domestic service, handwork for the nimblefingers and feeble brains of girlsind an appropriate educational alternative to thedifficult sums provided for boys.

Surrounding the history of embroidery is the story of how women's work wasdown-graded. deskilled, underpaid, an..1 exploited through processes of differ-entiation, marginalization, exclusion, segregation, and subordination (Holland,1991). As female employment moved from the home to the factor y. the sexualdivision of labour also transferred, ensuring that women remained hierarchicallysubordinate in the labour market. Within the work environment, gender hierarchyis maintained through technological advances; men maintain control over theprocesses and know-how while women operate the machinery with no idea ofhow it works. This process of gendering produces extremes whose immorality ismasked by their sheer stupidity In one fictory I visitAl while researching'Common Threads I a room 1611 of workers machined children's clothes swiftlyand accuratei y. maintaining parallel lines and curves on the small pieces of cloth.In the next room, other workers picked off the fluff that accumulated on machinesknitting socks. The union classification of the jobs were 'skilled' for the fluffpickers who were white men, and 'unskilled' for the machinists who were Asianwomen.

In the field of matlwmatics education, the seminal work of Walkerdine alla hercolleagues details the process :. whereby girls are 'gendeied out of achievement inmathematics (Walkerdine, 1989). The tinces that ease women and girls out of highachievement in mathematics, the forces that maintain mathematics as a maledomain, the fon es that stereotype particular activities as women's, and the forcesthat label those activities trivial and brainless are one mid the same. In my Own

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lary Harris

work, I have seen male mathematics teachers refuse to work with textiles in amathematics workshop because, as 'women's work', they found it degrading. Ihave female colleagues in mathematics education who refuse to do needlework,not because they do not like it, but because of the connotations it has as feminine.trivial and time-wasting. Both perspectives are equally prejudiced.

Public perceptions are hard to shift. It is difficult not to see things in agendered fashion when they are so prevalent in our culture. If the public perceivemathematics as a male subject in spite of the evidence to the contrary, then it willtake at least a generation of education to change it. As Willis (1989) points out,'Since we do not award prizes for low levels of achievement, boys' over-representation amongst the mathematically least able does not receive nwch mediaor educational coverage' (Willis, 1989, p. 14).

The damaging differences in perceptions as to what mathen ,atics education isabout, both inside and outside the profession. should be more tractable fromwithin. A largely undiscussed problem in mathematics education is that mathe-matics communicates mainly with itself. In England, for example, we still have notgrasped that the public's perception of nuthematics education, the one thatultimately has political power over the discipline, is totally different from its own(Harris, 1989). There have been programmes attempting to popularize mathe-matics, television programmes of mathematical fun and games, we have evenimproved participation rates and grades .n school. but v,-e have achieved very littlemovement amongst the power groups of government and industry who controleducation and the curriculum and who 'are raucously engaged in demanding adifferent scenario from the one we are offering' (Harris, 1989, p. 18). A siegementality within mathematics education is not the way to get things changed.

The greatest general effect of both 'Common Threads' exhibitions was thatthey communicated widely outside mathematics education as Well as inside it.There is no doubt that they did change perceptions. Their main message, however.was that the very stereotype that currently limits women in both mathematics andwork can be med as a weapon for their emancipation by the recognition.developmentind accreditation of the power tool that already exists within.

References

Ao M. (19861 Flypand Ilentao.: Histoly (!t 11'oinui iii Sornsc.fiont Antigua). ft, the LateNinchirith Ceram); London. Women's Press.

lit "1111, C.B. (198)) I !Iwo, y of Mathematio, 2nd ed.. New York. \k'iley.BR! I CI ti \ (19)3) ( :omnion Thtead m Botswana. Gabarono, British Council.Co( Kt Ilan W.H. (1982) Mathenlinhs Counts: Report 41. the Conmunee ot Inquay into (he

.\llthe,nUh in \,hools. 1 ondon. 11cr Majesty's Stationery ()niceP (1991) ihr Plulomiphy of .\lsuillenisumo Education. I ondon. balmer Press.

LI sit is. Is. M. (1985) 'Wrapping it up', II3, (1)ecember). pp. 44-5.111R R is. NI. 0989) 'Bash s'. .Ilathonam.s leathiv. 128 (September). p. 18.hititi. NI. (P)M 'Looking for the niaths ni work'. in 11N11111\ NI. (Ed)

Conunou Threads

MatImatics and 11,rk, London, Falmer Press, pp. 132-44.Hot I ANI), J. (1991) 'The gendering of work', in HARRIS, M. (Ed) Schools, Mathematics and

Work, London, Falmer Press, pp. 230-52.NAIR )tiAl COMMISSION ON El hl.ICA1 ION (1993) Learn* to Succeed: A Radical Look at

Education Thday and a Strategy /Or the Future, Report of the Paul Hamlyn NationalComnnssion on Education. London, Heinemann.

PARKI.11., R. (1984) The Subversive Stitch: Embroidery and the 3 lak* of the Fetninine, London,Women's Press.

WALKIADINE. V. (1989) Countiv Girls Out, London. Virago.Wu I AMS, R. (1961) Education and British Society in the Long Revolution, Harmondsworth,

Penguin Books.Wimi is, S. (1989) Gender and the Construction of Privilege, Geelong, Victoria, Australia.

Deakin University Press.

87

Part 2

The Cultural Context

The chapters in this part of the book provide a cross-cultural perspective on thediscussion of equity in mathematics education. We have grouped the chapters intwo sections according to whether they are based on comparative research or focuson particular issues within a single country The chapters in the first group exploredifferences either among several cultures or among several ethnic groups within asingle culture. The second group of chapters provide a perspective on the currentsituation in several countries across the world. Taken together, these paperschallenge monocultural assumptions underlying previous intervention approachesand question their suitability outside the culture in which they were developed.Several also question the relevancy, given local circumstances, of maintaining aconcern for equity in mathematics education.

A comparison of the cultural influences contributing to gender differences inlearning mathematics within different cultures is the focus of the chapter byGurcharn Singh Kaeley. He observes that the developing world witnesses a greaterdisparity in the education of both sexes than is apparent in the developed world,where gender differences favouring boys have by and large disappeared, at least inthe compulsory years of schooling. By contrast, he points to the situation in certainmatrilineal societies where females achieve the same or better mathematics resultsthan their male counterparts. Hawai'i is the only state in the US in which genderdifferences in mathematics favour girls. Paul Brandon, Cathie Jordan and TerryAnn Higa analyse the reasons for this phenomenon by examining the differentethnic groups within Hawai'i. Several factors are suggested, among them, differingadaptation to Hawai'i by male and by female immigrants, differential schoolcompliance, and the differential impact of peer culture among boys and girls ofnative Hawaiian ancestry. In the same vein. Sharleen Forbes examines genderdifferences in mathematics performance between the two different ethnic groupsin New Zealand. She reports that the average performance of Maori girls is lowerthan that of the Maori boys, whereas for girls and boys of European origin theaverage performance is the same. It appears that strategies to increase theparticipation and achievement of girls in mathematics in New Zealand have hada positive impact on girls of European descent, but have not met the needs ofMaori girls. This group of chapters concludes with a study, by Gilah Leder, of theinfluences of the print media on gender differences in learning mathematics.Focusing on print media in two ostensibly similar countries. Australia and Canada.Leder distinguishes among the media images using feminist and societal-psychological lenses. She concludes that subtle messages conveyed by the popular

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MC' Cultural Context

press are consistent with small but persistent difkrences in the ways females andmales perceive and value mathematics.

A strong challenge to monocultural perspectives on equity in mathematicseducation begins the second group of chapters in this part of the book. SalehaNaghmi Habibullah piesents a personal view of the issues in which she questionswhether equity in mathematics education is an appropriate goal in a developingcountry Even were it so, she challenges the assumption that western strategies forovercoming gender inequities would be equally effective in non-western societies.hi her opinion, there is a direct link between a country's socio-cultural conditionsand the manlier in which its people perceive and react to the issue of genderimbalance. Consequently, for many years to come, short-term goals and strategiesfor educational improvement will have to vary between countries according toeconomic conditions and socio-cultural realities.

The remaining chapters in this section describe the situation in a single countryand emphasize the influence of the cultural context on the learning of mathematics.Bermderjeet Kaur surveys the available empirical research documenting genderdifferences favouring boys in mathematics achievement and attitudes in Singapore.Nee la Sukthankar investigates the cultural issues affecting the women's education inPapua New Guinea. She describes the disadvantages girls endure as a result of lowenrolment at all school levels and attending poorly equipped schools. The mainfactor she cites as producing these disadvantages is family poverty which results inmost Cimilies being able to send only one child to a good school: because oftraditional gender-role expectations, this is usually a boy Other contributing factorswhich she explores are the loss oftraditional Papua New Guinean pre-mathematicalconcepts and language barriers to mathematics education.

Two chapters in this part of the book deal explicitly with the issue of single-sex mathematics education. Francoise Delon describes the impact on women'sparticipation in mathematics education after the recent desegregation of the mostrespected universities in France. Her account details the significant decrease thathas occurred in the numbers of female students studying mathematics and.ultimately, in the numbers of females pursuing careers in universities, schools andother prestigious professions. The former women-only learning environment hadbeen reassuring and stimulating to women and empowered many of them to entermixed competitive working environments common in the French society LikeDelon, Pat Hiddleston also advocates gender-segregated schooling, but in contrastto Delon, her account describes the situation in .1 very poor developing countryAccording to Hiddleston's research, girls in Malawi who attend girls-only schoolsachieve better results in the final school exannnations than those who attendcoeducationa schools and more of them enrol for 3 university education than girlsfrom coeducational schools. Furthermore, at the university level, although theyenter university with lower overall and lower mathematics qualifications thanmales. by the time they graduate, female students have higher results innuthematics and the sciences on average than male students.

90

Chapter 10

Culture, Gender and Mathematics

Gurcharn Singh Kaeley

Introduction

Recent meta-analyses of gender differences in mathematics performance indicatethat generalizations as to the superiority of either gender are impossible (Feingold,1989; Friedman, 1989; Hyde, Fennema and Lamon. 1989). The magnitude anddirection ofsuch differences depend on a number of factors. For ilistanceTe, typeof task, sample selection (Hyde. Fennema and Lamon, 1989), and the nature of theevaluation procedures (Kimball, 1989) all play a role. Female students appear todemonstrate superiority over their male counterparts on computational, mimer-icalind perceptual-speed tasks before high-school, with little or no differenceafter that time. However, on problem-solving and spatial tasks, male students seemto outperform females by the time they reach 13 years of age (Lummis andStevenson, 1990; Xu and Farrell, 1992). Boys achieve higher scores on standar-dized tests, as well as in the mathematics classroom (Kimball, 1989). QuotingAcademic Association of University Women, Streitmatter (1994) corroborates thatboth the National Association of Educational Progress and College Board data inthe US indicate that males outscore females on test items that assess highercognitive skills. Overall, the best one can conclude is that there appears to be amodest difference in mathematics pertbrmance in favour of male students.

Gender differences, such as those mentioned above, stein from culturalpressures and socialization processes characteristic of many countries in whichfemale students are not permitted to develop their mathematical abilities to theirfullest potential. Beliefs and prejudices about sex-appropriate behaviour arereflected in the expectations of parents, peers, school, and society (Leder, 1985).Such cultural stereotypes not only include the alleged natural superiority of boys'mathematical abilities, but also differences in beliefs about the utility of thediscipline for boys and girls. In maintaining these perspecti es, societies underminegirls' confidence and motivation, and impoverish their learning of mathenutics(Meece ci aL, 1982).

This chapter investigates the relationship between culture and genderdifferences in mathemati,-s enrolment and performance. In the pursuit of this goal,

1 C9 /

Gurdtarn Singh Kicky

four hypotheses are considered. First, the cultural norms in many developingcountries are responsible for producing enrolment disparities. Second, in thedeveloped world, cultural norms operate to discourage female students inmathematics to the point that their enrolment in mathematics courses declines assoon as enrolment in the subject becomes optional. Third, in societies where therole of women has changed, gender differences in mathematics performance arebeginning to decrease. Finally, in certain societies and cultural groups in whichwomen already have more power and authority females outperform males inmathematics.

Enrolment Disparities

Developed Countries

In North America, Europe, and many other developed countries there is universal,compulsory primary and secondary-school education provided through govern-ment taxation. In some of these regions, such as the UK, even tertiary educationis subsidized with students from disadvantaged backgrounds receiving financialassistance to pay for their books and other necessities.

Many researchers have studied enrolment patterns as well as performance inmathemazics courses in developed countries. There is consensus in their findingsthat female students enrol in smaller numbers than boys in the more advancedoptional high-school and college mathematics courses. Such reports derive froma wide variety of countries such as the UK (Shuard, 1982), the US (Fennema,1985), Australia (Leder, 1980) and a majority of other countries included in theSecond International Mathematics Study (Westbury, Russell and Travers, 1989).For example, in the US, the number of female citizens who earned a doctoraldegree in mathematics in the year 1986 to 1987 was only 20 per cent (NationalCenter for Education Statistics, 1992). More recently it has been noted in the USthat gender differences in participation are becoming negligible in all parts of thecurriculum except calculus wht.7V equal representation remains a problem(Streitmatter, 1994). In the UK, %chile the proportion of female students takingmathematics courses is increasing, it still remains disproportionately lower thanthat of males (Shuard, 1983). In the majority of the countries of the SecondInternational Mathematics Study, there are two or three times as many boys as girlsin advanced mathematics courses (Westbury. Russell and Travers, 1989).

Develop* Countries

Cultural norms also contribute to a deficit in female enrolment in institutions oflearning in nuny developing countries. Families which can afford to send only onechild to school will invariably send a boy,.since he is considered the future bread-wniner for the family. According to Anbesu and junge (1988)i survey of theGoljam region of Ethiopia showed that only 10 per cent of all children between

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Culture, Gender and Mathetnatks

the ages of 7 and 16 years were enrolled in school and, of these pupils, two-thirdswere boys. Similarly, Guy (1992) found that only 39 per cent of Papua NewGuinean secondary-school students were female. The literature from many otherdeveloping countries confirms this trend (Grace, 1991). More recently thePopulation Action International study reported that some eighty-five millionmore boys than girls receive elementary and secondary education throughout theworld (Chicago Tribune, 1994). These disparities become even more acute at thetertiary level (Kaeley, 1990). For example, citing a number of studies from PapuaNew Guinea, Kaeley writes that female students in the University of Papua NewGuinea in the years 1966, 1977, and 1988 comprised only 10 per cent, 15 per cent,and 18 per cent, respectively, of the total university population.

Mathematics Performance

Developed Countries

The First International Study of Achievement in Mathematics, which wasconducted in more than twelve developed countries, showed mixed resultsconcerning gender differences in mathematics performance (Thorndike, 1967).The achievement of boys was better than that of girls at age 13, particularly onverbal, as opposed to computational, problems. However, there was substantialvariation between countries, with gender differences being largest in Belgiuni andJapan, and least in the US and Sweden (Thorndike, 1967). Thorndike attributesthe small differences in the latter countries to the improved role of women in bothsocieties compared with Japan and Belgium.

To concentrate for the moment on the US, gender difkrences in mathematicsperformance have declined over the years since 1967 to the point where theirmagnitude is now very small (Hyde et al., 1990). In a meta-analysis of studiesconducted between 1960 and 1983. Feingold (1989) observed that genderdifferences in cognitive performance are disappearing, although the disparity at theupper levels of performance in high-school mathematics has remained constantthroughout the same period. These results were corroborated by Friedman (1989)in her meta-analysis of studies conducted between 1974 and mid-1987. Fur-thermore, she predicts that gender ditkrences in mathematics performance at alllevels of learning will vanish altogether in fiwty years.

From data compiled in the Second International Mathematics Study. Hanna,Kiindiger and Larouche (1990) compared sex differences in mathematics perform-ance of students in the final year of secondary school in fifteen north American,European and east Asian countries. They showed that differences were negligiblefor those countries where there were high levels of home support for both sexes,whereas countries where substantial sex differences were observed had lower levelsof home support. They suggested that since society has traditionally reinforcedboys' participation in mathematics, the degree of support provided by the homemay be a more crucial indicator of success in mathematics for girls than it is for

Gurcharn Singh Kaeley

boys (Hanna, Kiindiger and Larouche, 1990). This could mean that, when societalnorms are counter-balanced by family exnectations, gender differences disappear.Furthermore, the finding, that gender differences in mathematics performancevary between countries, runs counter to traditional arguments that attribute boys'superior mathematical performance to their biology, and suggests instead thatcultural factors offer a more viable explanation (Hanna, Kiindiger and Larouche,1990).

Developing Countries and Etimic Minorities in Developed Countries

The cultures of a number of ethnic minorities in the developed world are eitherfemale-dominated or gender-equitable (Driver, 1980). Even in cases where theindigenous culture is male-dominated, pressures by the dominant culture areproducing change. As the examples below illustrate, in some ethnic groups, thisresults in female performance in mathematics which is as good as, or better than,their male counterparts.

From a meta-analysis of 100 studies of gender differences in mathematicsperformance in the US, Hyde et al. (1990) demonstrated that there were no genderdifferences in the in-class mathematics performance of African-American andHispanic-American students. A slight difference for Asian-Americans was dis-covered and it fiwoured female students. Only for white students was there anyevidence of superior male performance, although the gap was small. In anotherstudy this time of Hawaiian students between the ages of 9 and 16 years, Brandon,Newton, and Hammond (1987) concluded that girls outperformed boys inmathematics in all the observed ethnic groups and that the female advantage wassignificantly larger in the Philippine. Hawaiian, and Japanese samples than it wasin the Caucasian sample (and this has continued, see also Brandon, Jordan andHiga. Chapter 11, this volume).

Another example of the influence of culture on mathematics performance isprovided by Geoffrey Driver (1980). He studied 2300 secondary-school graduatesof both sexes, includMg white students and students of West Indian descent, in fivemultiracial secondary schools in the U1. He observed that West Indian girlsoutpertbrmed West Indian boys in English language, mathematics, and sciencesubjects. White boys excelled over White girls, but not as markedly. Out of all fourgroups, the highest mean results were obtained by West Indian girls, with WestIndian boys doing less well than White boys and perfornnng at the level of theWhite girls. Driver argues that the superior performance of West Indian girls isprobably due to their culture. Citing a number of studies, he suggests that in ruralJamaica, the women rather than the men assume responsibility for the family'ssurvival. Upon immigration, West Indian funnies retain this custom and this isreflected in women's superior academic performance.

India is a developing country in which male dominance is the societal norm.In a survey of mathematics attainment in Indian schools, Kulkarni, Naidu, andArya (1969) observed inferior achievement by girls relative to boys in most states.Notwithstanding, there were sonic remarkable exceptions, for example, the

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1 u o


highly-developed Mangalore region of Mysore State where females outperformedmales.

The influence of culture on the role of women in society, on the cognitivedevelopment of children, and on the learning of mathematics can also be foundin Papua New Guinea and some South Pacific Island nations. In these countries,there are a number of matrilineal communities (see, for example, MelbourneUniversity Press and University of Papua New Guinea, 1972) in which womenhave more power and authority in the family than the men. Under suchcircumstances, girls arc treated with respect in the classroom as well as in thesociety as a whole. And indeed we find that females exhibit equal, if not better,performance in mathematics compared with males (Kaeley, 1990).

In 1983, Laney conducted a major research project on the cognitivedevelopment of primary-school children in Papua New Guinea. He observed thateven in patrilineal communities. there are no sex differences in cognitivedevelopment and the learning of mathematics. However, an analysis of thenationwide grade 6 (11-12 years of age) mathematics examination results does notsupport his conclusions, since boys' performance was found to be significantlybetter than that of girls (Measurement Services Unit, 1985). A study by Clarkson(1983), which focused on gender disparities in the learning of mathematics by15-year-olds in Papua New Guinea, also showed no differences in the perform-ance of boys and girls. An earlier study of a situation in which I 5-year-old boysexcelled over girls in mathematics noted that, given the right environment,achievement levels are equal (Silvey, 1978). At the tertiary level, Kaeley (1988) hasobserved that there were no gender differences in post-secondary mathematicsperformance at the University of Papua New Guinea. The reasons he suggestedfor this apparent discrepancy with the results of other studies are methodological:first, included in the sample were a number of female students from the matrilinealcommunities; second, as proportionally fewer females than males continue on tosecondary and post-secondary education, they are not representative of the femalepopulation of the country as a whole; and third, half the students in the samplewere engaged in distance education and thus were free from teacher-effect.Furthermore, he speculated that, since formal mathematics has only recently beenintroduced in Papua New Guinea, it was foreign to all students and thereforeimpervious to gender. Interestingly though. the only indigenous mathematicsdoctoral degree was earned by a woman.

Conclusion

Overall. it is evident that enrolment differences in developing countries predom-inantly fhvour male students. Even in the majority of developed countries, culturalinfluences encourage male students to take additional mathematics courses beyondthe stage at which they are compulsory. This situation, however, is slowly changingand gender differences in mathematics performance are decreasing as women's rolein society improves. There arc certain developing countries, as well as a number

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Gurcharn Singh Kiehl/

of ethnic communities within developed societies, in which women have morepower and authority than men. In these societies, either there are no apparentgender differences, or female students outperform their nule counterparts inmathematics. The main conclusion of this chapter is that culture is one of the mostimportant factors contributing to gender differences in the learning of mathe-matics.

References

ANIIIILI, B. and JUNGI., B. (1988) 'Problems in Primary School Participation andPerfbrmance in Bahir Dar Awr*'. (mimeograph), Addis Ababa, UNICEF andMinistry of Education.

BRANDON, P.R., Nt.w ION, B.J. and HAMM& ssoc, O.W. (1987) 'Children's mathematicsachievement in Hawai'i: Sex differences favouring girls'. American Educational ResearchJournal. 24, pp. 437-61.

Chicago li-ibune (1994) 'Study finds mak-female education gap'. in Chicago lribune, January31. Chicago, Tribune Publishing Company, p. 7.

Ci ARKM)N, P. (1983) 'Papua New: Guinea students' causal attributions for success andfailute in mathematics', Report No. 27, LaC, Mathematics Education Centre, PapuaNew Guinea University of Technology.

CoNNoRs. E.A. (1985) 'Report on the 1985 survey of new doctorates', Notices of theAmerican Mathematical Society. 32,6, pp. 768-71.

Diuvris, G. (1980) 'How West Indians do better at school (especially the girls)', in NewSociety, 51,902, pp. 111-14.

Ft rsaa )1 I), A. (1989) 'Cognitive gender differences arc disappearing'. American Psychologist.43,2, pp. 95-103.

Ft NNEMA, E. (Ed) (1985) 'Explaining sex-related differences in mAthematics: Theoreticalmodels', Educational Studies in Mathematics. 16,3, pp. 303-20.

FILII DMAN, L. (1989) 'Mathem,:tics and the gender gap: A meta-analysis of recent studieson sex differences in mathematical tasks'. Review of Educational Research, 59, 2.pp. 185-213.

GRA( 'I , M. (1991) 'Gender issues in distance education'. in EvANs, T.D. and KIM., B. (Eds)Beyond the Contemporary U ritini in Distance Education. Geelong, Deakin UniversityPress.

Guy, R.K. (1992) 'Privileging others and otherness in research in distance education', inEVANS, T.D. and PAR! is, J. (Eds) Research in Distance Education livo, (eelong, DeakinUniversity Press.

FIANNA, G.. KLINDk.i is, E. and LAita ii. 1111 , C. (1990) 'Mathematical achievement ofGrade 12 girls in fifteen countries', in Buis I ( IN, L. (Ed) Gender and Matlwinatics: AnInternational Nrspective. London, Cassell, pp. 87-97.

Hypi , j.S , Ft NNI MA. E. and LW( iN, S.J. (1990) 'Gender differences in mathematicsperform,nice: A meta,malysis', Psychological Bulletin. 1(17, 2, pp. 139-55.

KAI I I v, G.S. (1)88) 'Sex differences in the learning of post-secondary mathematics in aneo-literate society'. Educational Studies in Alathematics, 19, pp. 435-57.

y, G.S. (1990) Dist.mce Versus ,7ace-to-Face Learning: A Mathematics Test Case,

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Unpublished PhD Thesis, Waigani, University of Papua New Guinea.KIMBAI i., M.M. (1989) 'A new perspective on women's math achievement', Psythological

Bulletin, 105,2, pp. 198-214.KUIKARNI, S.S., NAIDU, C.A.S. and AMA, M.L. (1969) 'A survey of mathematics

achievement in Indian schools', Indian Educational Review 4,1, pp. 1-33.LANCY, D.E. (1983) Cross-cultural Studies in Cognithm and Mathematics. New York, Academic

Press.LEDER, G. (1980) 'Bright girls, mathematics and fear of success', Educational Studies in

.11athematics, 11,4, pp. 411-22.LEDER, G. (1985) 'Sex-related differences in mathematics: An overview', M FENNLMA, E.'

(Ed) 'Explaining sex related differences in mathematics: Theoretical models', Educa-tional Studies in Mathematics, 16,3, pp. 304-9.

Lummis. M. and STEVENSON, H.W. (1990) 'Gender differences in beliefs and achievement:A cross-cultural study', Developmental Psychoh)gy, 26, pp. 254-63.

MEANURIMEN r SI.Ity1(1.s UNI I (1985) Analysis if 1982 Grade 6 Examination Results,Waigani, Department of Education.

tcr, J.L., PARSON, J.E., KAC/AI A, C.M., GCB I, S.B. and Fiu HAMAN, R. (1982) 'Sexdifferences in mathematical achievement: Toward a model of academic choice' ,Psychological Bulletin, 91,2, pp. 324-48.

Mia UNIVERSE! PRY, ANE/ UNIVI.R.,I I y 01 PAPUA NM' GUINEA (1972)Encyclopedia of Ilvia Neu. Guinea, 2.

tioNI CFN Ellt i olt. El B.ICA I ION S IA1 mics (1992) Race/Ethnidty Trends in DqreesConferred by Institutions if. Higher Education: 1980-81 Through 1989-90, Washington,DC, US Chwernment Printing Office.

SI H.B. (1982) 'Differences in mathematical performance between boys and girls'.in CocKcitoil, WH. (Chairman) Mat1,ematic5 Counts, London. Her Majesty'sStationery Office.

Su %TY, J. (1978) Academic Success in PNG High Schools, ERU Research Report No. 26,Waigani. University of Papua New Guinea.

S IR1ii MAI I I It, J. (1994) Dward Gender Equity in the Classroom: L'eryday Twhers'and Practices, Albany, State University of New York Press.

TI R.L. (1967) 'Mathematics test and attitude inventory scores'. in 1-1t.'11 N, T.(Ed) International Studies of Achimement in Mathematics 11, Stockholm. Almqvist andWiksell, pp. 21-48.

WI.. I BURY, I., Rus,ali i . H.H. and TRAvl its. K.J. (1989) 'The content of the implementedmathematics curriculum', in TRAvl ItN, K.J. and Wi \IMAM I. (Eds) The HA Study ofAlathemarks 1: Analysis if Mathematks Curricula. Oxtbrd, Pergamon Press, pp. 167-202.

Xt.', J. and FARR! I 1, E. (1992) 'Mathematics performance of Shanghai high schoolstudents: A preliminary look at gender differences in another culture', School Scienceand Matheminks, 92, pp. 442-5.

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= -

Chapter 11

Why Hawan Girls Outperform Hawai'iBoys: The Influence of Immigration andPeer Culture

Paul Brandon, CathieJordan and Terry Ann Higa

Gender 0.: Terences in mathematics achievement in the United States historicallyhave favoured boys. In the State of Hawai'i, however, these differences havefivoured girls. What is different about Hawai'i that might help account for thestate's unique pattern of gender differences in mathematics achievement?

In this chapter, we suggest that some characteristics of HawaiTs immigrant andnative Hawaiian populations may have enhanced girls' educational achievement andnegatively afkcted boys' achievement. First, we give a brief summary ofthe findingson gender differences in mathematics achievement across the United States and inHawai'i and discuss gender differences in verbal skills among children nationwideand in Hawafi. Second, we summarize research on the effects of immigration onmales and females and present the thesis that, in Hawai'i, such effects may havemanifested themselves in Hawaiis gender differences in educational achievement.Third, we summarize anthropological theory on schnol compliance and ethno-graphic data on peer culture among native Hawaiian youngsters, and we suggest howthese are applicable to gender differences among native Hawaiians:

Gender Differences

Maccoby and Jacklin, in their landmark survey of research on gender differencesin nuthematics achievement, concluded that American boys and girls were 'similarin their early acquisition of quantitative concepts, and their mastery of arithmeticduring the grade-school years' (Maccoby and Jacklin, 1974, p. 352), but that boys'mathematical skills increased faster than girls' after about 12 years of age. Recentmeta-analyses have shown that differences in mathe-natics achievement havedecreased and largely been eliminated (Friedman, 1989: Hyde, Fennema, andLamon, 1990) except among high-school and post-secondary students on somemathematics subtests, on which boys have continued to show somewhat higherlevels of achievement than girls.

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1

14,4 Hawari Girls Outpey-orm Hawai'i Boys

In Hawai'i, studies have consistently shown elementary- and secondary-school girls outperforming boys on mathematics tests (Brenner, 1984; Holmes,1968; Kamehameha Schools/Bishop Estate, 1983; Marshall, 1927; Stewart, Doleand Harris, 1967). Brandon, Newton and Hammond (1987) reported an analysisof 1982 and 1983 Stanford Achievement Test results from HawaiTs statewidetesting programme. For children of Filipino, native Hawaiian, Japanese, and whiteancestry, results on three mathematics subtests for Grades 4, 6, and 8 (ages 9-14years) and one subtest for Grade 10 (ages 15-16 years) showed that girlsoutperformed boys in seventy-six of eighty comparisons. Results across ethnicgroups, subtests, and testing years were combined and effect sizes (that is, the girls'mean minus the boys' mean divided by the standard deviation of both groupscombined) were calculated. The effect sizes favoured girls and were greater amongolder children than among younger children, with sizes ranging from 0.12 forGrade 4 to 0.26 for Grade 10. A finding that is particularly relevant for this chapteris that differences favouring girls were greater among children of non-Whiteancestry than among White children.

Brandon and Jordan (1994) have expanded on these findings in two ways. Firstthey analysed 1991 Hawafi achievement test results, showing that genderdifferences in mathematics achievement among Hawai'i school children havecontinued. Second they compared findings for Hawafi on the 1990 NationalAssessment of Educational Progress state-level eighth-grade mathematics assess-ment with the findings of thirty-six other states, the District of Columbia, and twoterrilories. 1-lawai'i was the only one of the forty participating jurisdictions inwhich girls' total-test mean scores were significantly higher than those of the boys.Therefore, Brandon znd Jordan concluded that the Hawai'i gender differences areunique among the American states.

Another arc in which boys' and girls' achievement has traditionally andwidely been compared is verbal skills. In a meta-analysis of studies on differencesin verbal skills, Hyde and Linn (1988) found an effect size of 0.11. They concludedthat 'overall, studies of gender differences in verbal ability support the conclusionthat these differences are now negligible' (Linn and Hyde, 1989, p. 18). In contrast,analyses conducted for this chapter show that Hawafi Stanford Achievement Teststatewide testing results for four grades in reading in 1991 show effect sizes rangingfrom 0.12 to 0.24, with a mean effect size of 0.18. Thus, the results for Hawafisuggest larger differences favouring girls than the national results, and, togetherwith the results found on gender differences in mathematics achievement, suggesta general educational achievement advantage for Hawai'i girls relative to boys.

What explanations for Hawafi educational gender differences might we gleanfrom the literature? In the next section, we suggest that ininngration might haveaffected Hawai'i boys' and girls' educational achievement and attainment levels.

The Impact of Immigration

'Legal immigration to the United States ... has been dominated by females for thepast half century' (Pedraza, 1991, p. 304), yet few studies have focused on women

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Paul Brandon, CathieJordan and Terry Ann Higa

immigrants (Pedraza-Bailey, 1990). Nevertheless, by examining reports of studieson immigrants published in the psychological, sociological, and anthropologicalliterature, we have found sufficient literature to support the claim that a theme hasemerged: inmiigration into western nations may be more beneficial for femalesthan for males, at least in the short run, with the consequence that femaleimmigrants experience greater educational success than male immigrants.

Female hnmirants May Adapt More Easily than Males

Studies of Jewish, Irish, and Caribbean immigrants to the US and England havesuggested that female immigrants experience less difficulty than males in adaptingto their new lives. Summarizing the status of Jewish working-class maleimmigrants in the late-nineteenth and early-twentieth centuries, Hertzberg saidthe 'evidence of the destruction of the role of the flither is overwhelming'(Hertzberg, 1989, p. 196). Less than 20 per cent of the men during this periodearned sufficient income to support their families, and many deserted theirfunilies, resulting in the development (or, in Hertzberg's words, the 'invention')of the 'Jewish mother' as the family's protector (Hertzberg, 1989, p. 196). AmongIrish immigrants, men experienced a similar 'decline in status and power withintheir families as a result of migration, pushing women wives and mothersinto authoritative roles far greater than they had experienced' in Ireland (Diner,1983, p. 46). Migration was a liberating" experience for Irish women. Theyconsciously had chosen to leave a society that offered them little as women toembrace one that proffered to them greater opportunity ...' (Diner, 1983, p. 140).Pessar (1986) found that women inunigrants to the US from the DominicanRepublic were reluctant to return to the Dominican Republic because of the gainsthey had experienced after migrating. As Pessar said about these womenimmigrants, 'in contrast to men, migration does not rupture the social sphere inwhich women arc self-actualized' (Pessar, 1986, p. 276). According to Foner(1978), among Jamaican immigrants to England, immigration was more positivefor women than men because the women had become less dependent upon men.

Studies of Chinese immigrants have shown similar post-immigration malefemale differences. In a study of Chinese Americans, Yu (1984) reported that therelationship between life satisfiction and acculturation was stronger for Chinese-American females than miles. Huang compared male and female Chinese-American students and reported that females 'prove to be more sensitive toWesternization and more balanced in their acculturation process' (Huang, 1956,p. 25) than males. In an anthropological study of first-generation ChineseAmericans' dating behaviour, Weiss (1970) reported that females had adjustedmore quickly and easily than males to American social customs, that they weremore accepted by Whites as dating partners and possible mates, and that they oftenthought that Chinese-American males were inept dating partners.

In studies of Asian immigrants living in Hawai'i, similar patterns suggestingless difficult adjustment among !emales than males have been found. Arkoff,Meredith and Iwahara (1962) concluded that male Japanese immigrants might

100

4.1

14/7iy Hawai'i Girls OutpeyOrm Hawai'i Boys

have accultnrated less easily or quickly to American culture than female JapaneseinuMgrants. Among later generations of Japanese Americans in Hawai'i, findingson gender differences suggest that the effects of differential adaptation havepersisted. For example, Meredith and Meredith (1973) compared JapaneseAmericans and White Americans and showed that Japanese-American males intheir study were more introverted than the White males but that no suchdifferences were found between the Japanese-American and White females.Similarly, Bartos and Kalish (1961) showed that among college-campus leaders, theproportion of Japanese-American females was almost three times the proportionofJapanese-American males.

Immigration May Affiyt Educational Achievement and Attainment Levels

Published evidence for gender differences in immigrants' educational achievementis sparse and presents a mixed picture. A summary of late-nineteenth and early-twentieth century immigrants to the US (Olneck and Lazerson, 1974) showed thatimmigrant girls in most ethnic groups had completed more years of high-schoolthan immigrant boys, but among Russian Jews and southern Italians, boys hadcompleted more years of school than girls. Brandon (1991) analysed data from the1986 follow-up of the national 'High School and Beyond' survey. He concludedthat Asian-American females have reached high levels of educational attainmentmore quickly than Asian-American males and that these differences have beenlargest among Chinese Americans, Filipino Americans. and recent immigrants ortheir children. Duran and Weffer, however, examined educational data onMexican-American immigrants and found that 'girls had lower achievementbefore high school and the differential increased through the high school years'(Duran and Weffer. 1992, p. 177).

Such findings suggest that group-specific cultural attitudes toward theeducation of the sexes may have influenced inulligrants' educational achievementor attainment levels. Among late-nineteenth and early-twentieth century immi-grant Russian Jews, for example, the education of males was highly valued, andamong immigrant southern Italians during the same period, the education offemales was disparaged. Campbell and Connolly (1987) found that parents'expectations of White American students' secondary and post-secondary per-formance were greater for boys than for girls. Among Asian Americans, however,Campbell and Cor golly found that parents' expectations were the same for bothsexes. Among pre-colonial Filipinos, men and women were regarded in anegalitarian manner, and today the education of Filipino women is highly valued(Alcantara, 1990). The effects of these attitudes toward Filipino women'seducation are manifested in the high educational attainment rates among Filipinowomen in the Philippines, where 'the literacy rate among women is equal to thatof men' (Alcantara, 1990). They are also manifested in the educational attainmentrates of young Filipino immigrant women. In Table 11.1, we show analyses of datataken from the US Census Bureau's computer runs of unpublished 1980 censusresults (US Bureau of the Census, 1986) on the educational attainment of male and

101

Paul Brandon, CathieJordan and Ti'rry i4 n

Table 11.1: Percentages of native-born and immigrant Asian-Amencan males and females aged20-9 years in 1980 who had graduated from college

Native-born Immigrants

Male. _...

Female Male...___ __ _ .

Female

Filipinos 10.8 14 8 23 1 35.3Chinese 40.3 40.5 36 8 31..6

Vietnamese 15.7 20.0 9.8 6 0Japanese 31.2 34 5 31.5 23.7Koreans 23.5 21.9 23.7 15 0Asian Indians 28 2 22.5 53.8 43.3

Source: US Bureau of the Census (19861

female immigrant and native-born Asian Americans aged 20-9 years in six ethnicgroups. As can he seen, gender differences favouring female immigrants are foundonly among Filipinos.

Immigration and Educational Achievement

Hawaii is the youngest American state, and much of its population is comprisedof the most recent generations of immigrants to the US. Before the mid-1800s,when many European immigrants had already entered the US mainland, onlysmall numbers of non-native people had come to Hawaii to stay. From 1852 to1930, however, about 400 000 immigrants were brought to the state to work onplantations (Fuchs, 1961). In recent years, Hawai'i has continued as a home fornew immigrants; for example, Hawaii's 1987 population comprised only 0.44 percent of the total US population, but 1.13 per cent of all immigrants to the US in1987 chose Hawai'i as their intended residence (National Technical InformationService, 1990). Most recently, over half of these immigrants have conic from f,iePhilippines (Caces, 1985; Department of Business and Economic Development,1988).

If immigration affects females more positively than males, the relatively highproportions of immigrants in Hawaii throughout its recent history might accountat least in part for its unique gender differences favouring girls in mathematicsachievement. The argument, supported by the findings and studies summarized inthis chapter so far, goes as follows. First, gender differences in Hawaii arc foundin both mathematics and verbal skills, suggesting a general educational achieve-ment advantage for girls. Second, these differences are greater among non--Whitesthan among Whites, leading us to focus in Hawaii on Asian immigrants, thepredominant non-White immigrant group. Third, among several national orethnic groups nationwide (including Asians), immigration may have been morebeneficial for women than for men, at least in the short run. Fourth, the positiveeffect of immigration on females in Hawaii has been maintained beyond the firstgeneration and has been seen in immigrants' descendants. Finally, Filipinos, who

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Why Hawin'i Girls Outpey.orm Hawai'l Boys

comprise Hawal'i's largest recent immigrant group, have high regard for theeducation of females, thus reinforcing long-standing Hawal'i trends favouringfemales' outperformance of boys in school. This evidence forms the foundationfor our suggestion that the effects of immigration help account for genderdifferences in educational achievement favouring girls in Hawai'i.

School Compliance and Native Hawaiian Peer Culture

Rationales for School Compliance

Anthropologists and sociologists have pointed out that schools require children togive up independence and expect them to acknowledge the authority of strangerswho have no pre-established legitimacy. All children have to decide whether theyare going to comply with what schools and teachers ask of them. Especially forminority culture children, much of what the teacher and school require may seemstrange and even sometimes offensive. Since sonie minimal degree of studentcompliance is a necessary condition for success in school, the decision aboutwhether to comply with school rules and teacher requirements is a crucial one.

John D'Amato (1986, 1993) has developed a useful perspective on schoolcompliance, based mainly on his work with primary-grade native Hawaiianchildren. According to D'Amato, there are two kinds of rationales for complyingwith the teacher's wishes, 'structural' and 'situational'. A structural rationale forcompliance involves knowledge of, and belief in, external advantages of complyingthat are of such magnitude that children are willing to tolerate even very punishingconditions internal to the school context in service of external goals. A structuralrationale for compliance could involve, for example, the conviction that success inschool leads to success in life, or parental and familial pressure on the student tocomply with school requirements and excel in school, even at the sacrifice of otherculturally important roles and values.

A situational rationale for compliance, on the other hand, involves factorsintrinsic to the school or classroom. Children who do not have a strong structuralrationale for complying with school demands may nevertheless comply if doing sobrings them valued payoffs within the school setting and does not violate theirsense of identity or their beliefs about appropriate behaviour. In particular, at leastwith native Hawaiian children (D'Amato, 1993), when classroom practice and theteacher's claim to authority do not conflict with the structure and demands of thechildren's peer groups, then even children who have little in the way of structuralrationales for compliance may cooperate with the teacher.

Peer Groups and Rationalesftr School Compliance

How does this relate to malefemale differences in the educational achievementscores of native Hawaiian students? We suggest that such gender difkrences mayoccur because of differences in the availability of structural and situational

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Paul Brandon, CathieJordan and Thry Ann Higa

rationales for compliance and achievement.Hawaiian fanulies and kinship networks traditionally were generationally

organized; many still tend toward this kind of organization (D'Amato, 1986;Howard, 1974). One consequence of this organization is that adults interact mostlywith adults, and children with other children. After infancy, children are socializedto become part of an interdependent, interacting group of siblings which bearscollective responsibility for much of the work of the household, includingconsiderable childcare (Gallimore. Boggs, and Jordan, 1974; Jordan, 1981). Thissibling group is central to the lives of many Hawaiian families. In the sibling group,an older child, usually the oldest girl, often acts as helper or 'right hand' to hermother (D'Amato, 1986; Gallimore, Boggs and Jordan, 1974). She is responsiblein a special way for looking out for her younger siblings and for seeing that thework of the sibling group gets done. She has a special place in the affections of hersiblings, as well as a special degree of influence with them. In addition, because ofher relationship with her mother, she is a link between the generations.

Starting in later childhood, and becoming increasingly important in theteenage years, companion groups of close-age, same-sex peers become important,especially for boys (Gallimore, Boggs and Jordan, 1974). The peer companiongroup tends to remain central for males into early adulthood, even after familyformation and marriage. It exerts a powerful 'pull' for boys and young men whichis only gradually overcome by the needs and ties of the family and kinship network.In these same years, females are becoming increasingly involved in kin and flimilyconcerns. Kinship networks tend to be held together by ties between females, withmales being more peripheral (D'Amato, 1986). Because of the importance oflinksamong female kin, young women may be more involved in cross-generationalinteractions than young men are.

In school, there is evidence of a strong peer orientation among both boys andgirls (Gallimore. Boggs and Jordan, 1974). In D'Amato's (1986) study of a class oflow-income, elementary-school native Hawaiian children, girls and boys hadseparate, differently organized peer groups with different kinds of objectives. Bothboys' and girls' peer groups operated in a dynamic of rivalry in which individualsstrove to establish and maintain themselves in positions of equality or near equalitywith other members of their peer group. However, the boys in the study engagedin rivalry for dominance among themselves, while the girls engaged in rivalry forcentral social positions within their peer group. Also, for girls, closeness to theteacher seemed to be an important counter in their game of centrality, in a waythat did not have a parallel in the boys' dominance ganw. In addition, in sonwclassrooms, one girl appears to take on the position of chief helper to the teacher,echoing the role of mother's helper. In D'Amato's study, this was the same girlwho occupied the most central position in the peer group. This girl thus crossedthe generational line, just as the mother's helper does in the finnily, and may haveprovided a link between the teacher and the girls' peer group that did not exist forboys.

These patterns in children's lives at home and at school may, at least in part,be responsible for gender differences in achievement among native Hawaiian

104I. _A- L'

Hawai'i Girls Outped-orm Hawai'i Boys

youngsters. For Hawaiians, as a group, there is little structural rationale forcompliance and achievement in school. With a history of conquest andcolonization by the United States, they are at the lower end of the economicspectrum. They have suffered long-term discrimination against their culture andlanguage by the dominant group. This kind of experience does not inspire muchtrust in schools as an institution of the dominant society, or faith in schooling asa realistic road to success in life.

That being the case, we need to look to the classroom setting and thedynamics of interaction in school to determine the relative strength of situationalrationales for (or against) school compliance. Since peer orientation among nativeHawaiian children is strong and the peer group important its influence must beconsidered cuefully. We suggest that the way girls' peer groups operate is morecompatible with the ordinary school situation than is the organization andoperation of boys' peer groups. It thus may constitute a kind of situational rationalefor compliance that does not exist to the same degree for boys. Among youngerchildren, the dominance rivalries and struggles of the boys may present moreserious and disruptive challenges to teachers and more difficult barriers tocompliance than does the centrality-oriented rivalry of girls. This is especially truesince girls can actually incorporate closeness to the teacher into their centralitygame, and since the central girl in the peer group can also serve as a facilitator andbridge to friendly relations with the teacher. Among older students, it is possiblethat, as girls' concerns and energies turn more toward family obligations and kinties, the power of their school-based peer groups may diminish somewhat, but thatit remains as strong or stronger for older boys as it is for younger boys. The valuesand interactions of the peer group may have much more appeal for boys thananything the traditionally organized school usually offers in the way of situationalrationales for compliance.

Peer Group Culture and Other Ethnic Groups

The importance of the male peer group and its rivalry games seems to haveprecedents in native Hawaiian culture and to appear most dramatically in schoolswith high native Hawaiian populations, but to a certain extent, it is also aphenomenon of HawaiTs 'local' youth culture as a whole. Thus, it may affect allof HawaiTs male students to a greater or lesser degree. and we can speculate abouthow this f-act may relate to the relative sizes of gender differences in achievementscores for other ethnic groups.

Filipino children are more likely than Japanese or Caucasian children to livein the same neighbourhoods and attend the same schools as native Hawaiianyoungsters. Therefore, thctors similar to those associated with the male peer-grouporganization for Hawaiians may have more effect on Filipino boys and less effecton Japanese and Caucasian boys, resulting in a larger differential between Filipinoboys and girls' achievement.

Achievement-test effect sizes fiwouring girls may be the smallest for Caucasian

1 1 4

105

Paul Brandon, CathieJordan e Terry Ann Higa

children because many are immigrants from the mainland or, as the children ofmilitary families, are in Hawai'i for only relatively short periods of time. Thus,Caucasian students air perhaps the least likely to participate in the local schoolculture, including the peer-group organization of boys with its attendant potentialfor impacting the degree of school compliance.

Conclusion

In this chapter, we have summarized literature suggesting that ditTerentialadaptation to their new homeland by male and female immigrants to the state ofHawai'i and differential school compliance among native Hawaiian boys and girlsmay help account for gender differences favouring Hawai'i girls on bothmathematics and verbal-achievement tests. The etTects of immigration were seenin the state as early as the 1920s and, in recent years, may have continued becauseof the high proportion of Filipino immigrants to the state. We speculate thatdifferential school compliance among native Hawaiian boys and girls may also helpaccount for gender differences in achievement among boys and girls of the otherethnic groups in the state. Of course, in addition to immigration and schoolcompliance, there may very well be other sociological, anthropological, psycho-logical, or historical factors which also affixt gender differences in Hawai'i. Theseintriguing differences in achievement, unique among the American states, deservefurther examination by educational researchers, practitioners. and policy makers.

References

Ai CAN IARA, A.N. (1990) Gender Differentiation: Public versus Private Power in FamilyDecision Making in the Philippines, Unpublished doctoral dissertation. University ofHawai'i. Honolulu.

AR KOI I, A., MI.R1.1/1 I ii, G. and IWAIIAILA, S. (1962) Dominance-deference patterning inmotherland-Japanese, Japanese-Americanind Caucasian-American students', Journaltf Social Psychology, 52, pp. 368-42ft

BAR O.J. and KAI 'SI!, R.A. (1961) 'Sociological correlates ot' student leadership inHawaiT,Journal (f Educational Sodology, 35. pp. 65-72.

BRAND( )N, P.R. (1991) 'Gender differences in young Asian Americans' educationalattainmentS'ex Rolo, 25, pp. 47-63.

LIRANI)oN, P.R. and JORDAN, C. (1994) *Gender differences favouring Hawai'i girls inmathematics achievement: Recent findings and hypotheses', Zentralblatt .fur Didaktikder Alathetnatik, 26, 1, pp. 18-21.

BISANIK)N, P.R., Ni w o)N, B.J. and HmarvioNIS O.W. (1987) 'Children's mathematicsachievement in Hawai'i: Sex differences favouring girls'-Itnerkan Educational Researchfollow!, 24, pp. 437-61.

131k NNI Is, M.E. (1984) Standardized Arithmetic Testing at KEEP (1975, 1977). Unpub-lished manuscript. Kamehameha Schools/Bishop Estate, Center for Development ofEarly Education, Honolulu.s, M.E (1985) Personal Networks and the Material Adaptation of Recent Immigrants:

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Why Hawai'i Girls Outperfirm Halvai'i Boys

A Study of Filipinos in Hawai'i. Unpublished doctoral dissertation. University ofHawai'i, Honolulu.

CAMPBEI I , IR. and CONNOI I Y. C. (1987) Deciphering the effects olsocializationjournalof Educational Equity and Leadership, 7, pp. 208-22.

D'AMATO, J. (1986) 'We Cool, Tha's Why.': A Study of Personhood and Place in a Classof Hawaiian Second Graders, Unpublished doctora; dissertation. University ofHawail, Honolulu.

D'Ammo, J. (1993) 'Resistance and compliance in minority classrooms', in JACOB, E. andJORDAN, C. (Eds) Minority Education: Anthropological Perspectives, Norwood, NJ, Ablex.pp. 104-10.

DITARTMEN I (V BUSINFSS AND ECONOMIC DEVLI OPME.N r (1988) The State of Hawai'iData Book, Honolulu.

DMA, H.R. (1983) Erin's Daughters in America: Irish Immigrant libmen in the NineteenthCentury, Baltimore, John Hopkins University Press.

DURAN. B.J. and Wm-i u. R.E. (1992) 'Immigrants' aspirations, high school processindacademic outcomes', American Educational ResearchJournal, 29,1. pp. 163-81.

FONFIL, N. (1978) Jamaica Farewell: Jamaican Migrants in London, Berkeley University ofCalifornia Press.

FiummAN.L. (1989) 'Mathematics and the gender gap: A meta-analysis of recent studieson sex differences in mathematical tasks'. Review of Edmational Research. 59.pp. 185-213.

Ftvi is, L.H. (1961) Hawaii Pono: A Social History. New York, Harcourt and Brace.GA1 I N1( R., BoGcs, J. and JOILI %AN, C. (1974) Culture, Behatior and Education: A Study

Hawaiian-Atnericans. Beverly Hills. CA, Sage.Hi si /to ik.c;, A. (1989) TheJews in America, New York. Simon and Schuster.

s. G.C. (1968) A Study of Value and Attitudinal Correlates of School Achievementand Success in Nnakuli, Unpublished doctoral dissertation. Syracuse University,Syracuse, NY.

HowARi ), A. (1974) Ain't No Big Thing: Coping Strategies in a Hawaiian-AmericanComtmmity, Honolulu. University Press of Hawail.

HUANG, L. (1956) 'Dating and courtship innovations of Chinese students in America',Marriqe and Family Living. 18, pp. 25-9.

HYDI J.S., FI,NNMA, E. and LAMON. S.J. (1990) 'Gender differences in mathematicsperformance: A meta-analysis', Psychological Bulletin, 107, pp. 139-55.

J.S. and LINN, M.C. (1988) 'Gender differences in verbal ability: A incta-analysis',Psychological Bulletin, 104. pp. 53-69.

JoRDAN, C. (1981) Educationally Effective Ethnology: A Study of the Contributions ofCultural Knowledge to Effective Education for Minority Children, Occasional paperof the Center for Development of Early Education, Honolulu, Kamehameha Schools/Bishop Estate.

KAMI I !AM IA SCI R SCSI s/1.31 \IR )1' Es IAI I (1983) Native Hawaiian Educational AssessmentProject Final Report, Honolulu.

LINN, M.C. and HYDF, J.S. (198.)) 'Gender. mathematics. and science', EducationalResearcher, 18,8, pp. 17-19,22-7.

MACCt my, E.E. and JActo IN, C.N. (19i t) The Psychology of Sex Ditkrences, Stanford,Stanford University Press.

MARS! IA; i , E.L. (1927) A Study of the Achievement of Chinese and Japanese Children inthe Public Schools of Honolulu, Unpublished master's thesis, University of Hawai'i.Honolulu.

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MEREI MTH, G.M. and MERFAwr11, C.G.W. (1973) 'Acculturation and personality amongJapanese-American college students in Hawaii', in Sul:. S.S. and WAGNER, N.N. (Eds)Asian-Americans: Psychological Perspectives, Palo Alto, CA, Science and Behavior Books,pp. 104-10.

NAHONAL TECHNICAL INFORMATION SERvICE (1990) 1989 Statistical Yearbook of theInmugration and Naturalization Service, Washington, DC, US Department of Com-merce.

OLNECK, M.R. and LA/ERSON, M. (1974) 'The school achievement of immigrantchildren: 1900-1930'. History of Education Quarterly, 14, pp. 453-82.

PI I MA1A, S. (1991) 'Women and migration: The social consequences of gender', AnnualReview of Sociology, 17, pp. 303-25.

PEDRAZA-BAITEY, S. (1990) 'Immigration research: A conceptual map', Social ScienwHistory, 14,1, pp. 43-67.

PESAlk, P.R. (1986) 'The role of gender in Dominican settlement in the United States',iii NASH, J. and SAFA, H. (Eds) Mmen and Change in Latin America, South Hadley, MA,Bergin and Garvey, pp. 273-93.

STEWARI, L.H., Doi i, A.A. and HARRIS. Y.Y. (1967) 'Cultural differences in abilitiesduring high school', American Edwational Research Journal, 4, pp. 19-29.

US BiiRilAu o H IF CENSUS (1986) 'Numbers of years of school completed, by immigrantstatus, age groupind ethnic group', Unpublished computer-generated tables.

Wuss, M. (1970) 'Selective acculturation and the dating process: The patterning ofChinese-Caucasian interracial dating'. Journal of Marriage and the Rimily, 32,

pp. 273-8.Yu, L.C. (1984) 'Acculturation and stress within Chinese American Journal of

Comparative Family Studies, 15, pp. 77-94.

108

Chapter 12

Mathematics and New ZealandMaori Girls

Sharleen D. Forbes

Introduction

New Zealand was one of a number of countries that participated in the 1981 SIMS(Second International Mathematics Study) mathematics survey In certain areas ofmathematics, such as measurement and arithmetic, New Zealand students failed toattain a high international ranking (Department of Education, 1987). Moreover,small but statistically significant gender differences in overall mathematics perform-ance were observed as early as Form 3 (the first year of secondary school) (Wily,1986). In comparing ethnic groups within New Zealand, marked differenceswhich, in general, increase throughout the Form 3 year (Garden, 1984) werenoted between Maori (indigenous New Zealanders) and other students. Data forthe next international survey the Third International Mathematics and ScienceStudy is not due to be collected in New Zealand untillate 1994. A smaller nationalstudy was therefore commissioned by the Ministry of Education and undertakenby the women's collective, Equity in Mathematics Education, using 1989 data toupdate some of the SIMS results.

The Equity in Mathematics Education study involved a 9 per cent sample ofall first-year secondary students in one of the four education regions that compriseNew Zealand. While Maori students comprised only 17 per cent of the first-yearsecondary school population in 1989 (Ministry of Education, 1990), they wereslightly over-represented (23 per cent) in the sample. In March, the beginning ofthe school year in New Zealand, a sample of 1639 students completedquestionnaires focusing on their family background and attitudes towards mathe-matics. The sample was designed in such a way that approximately half of thestudents were from 'intact classes', that is a class consisting of all students taughtmathematics together at the same time and by the same teacher. Tests from theSIMS survey were administered to students in Marchmd again at the end of theschool year in November, to permit an assessment of the impact on students of themathematics programme. The purpose of this chapter is to investigate themathematics achievement of Maori girls, and to determine whether the observed

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Sharleen D. Forbes

overall gender differences in mathematics achievement in New Zealand areculture-specific. Since the average age of the students in the study was 13 years,this report provides an analysis of the attitudes, expectations and mathematicaloutcomes of 13-year-old Maori girls. For full details of the study including acomparison of the overall, topic, and question-specific results between 1989 and1981, and an analysis of gender and ethnic differences in attitudes towards, andperformance in, mathematics, see Forbes et al. (1990).

Backgrounds and Attitudes of Maori Students

Under the Treaty of Waitangi, the Maori people have special rights with respectto their land, fisheries, and forests. However, since signing the agreement in 1840,they have never experienc,fd the same social and Pconomic advantages enjoyed byother sectors of New Zealand society.

Maori is not the first language of most Maori children. While approximately43 per cent of the 384 Maori students in the study said that their parents normallyspoke the Maori language at home, only 10 per cent of the students claimed tobe able to speak it themselves. Perhaps as a result of deliberate attempts byeducators to 'assimilate' Maori students, English has become their dominantlanguage and is used aimcst. exclusively in everyday conversation. The preservationof the Maori language is urrently a major concern in New Zealand, with manyMaori parents beginning to demand bilingual or immersion Maori education fortheir children (Davies and Iicholl, 1993).

The occupations of employed parents were classified using the Elley-Irving(Johnson, 1985) scale for males and the Irving-Elley (1977) scale for females. The'unskilled' category includes manual labourers, but not home-makers, theunemployed, or those on welfare. As Table 12.1 shows, parents of Maori studentsare much more likely than parents of European descent to be in 'unskilled' or'semi-skilled' occupations (categories 1 and 2 in Table 12.1).

Maori parents were also more likely than their European counterparts to haveno education beyond the secondary level (Table 12.2). Less than 20 per cent ofboth Maori mothers and M -)ri fathers participated in tertiary education,compared with approximately one-third of parents of European descent. Therewere reasonably high positive correlations between the employment and educationvariables (p < 0.05), indicating that a parent with limited education would mostlikely be in an unskilled occupation, with a spouse in a similar position. Not onlydo most Maori children come from low-income families, but as already noted,their parents are also likely to possess only a limited education. This might beexpected to affect the attitudes of Maori children towards higher education, as wellas to hinder their participation in certain subject areas such as mathematics.

The disadvantages experienced by Maori students in terms of their low socio-economic status, coupled with the discontinuity they experience between theirhome and school cultures, influence their attitudes towards post-secondaryeducation. Only 32 per cent of the Form 3 Maori students expected to continue

110 4 1

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Mathematics and Nen, Zealand Maori Girls

Table 12.1: The percentage of parents of European and Maori origin in each socio-economiccategory

Fathers Mothers. .

European Maori European Maori

1 7 23 16 272 12 34 10 183 34 24 25 224 23 14 33 245 12 9 1 1 76 12 3 5 2

Note Scale 1 = Unskilled, 6 = Professional

Table 12.2. Highest level of education of parents (by percentage)

Fathers Mothers


Primary only 2 5 2 5Secondary only 63 80 67 77Tertiary 35 15 31 18

with formal education beyond secondary school, compared with 57 per cent ofstudents from European backgrounds. There was a significant correlation(p < 0.05) between the students' expected number of years of further educationand the parents' education levels or occupations, indicating that these lowexpectations are, at least in part. influenced by home characteristics.

The March questionnaire asked students to indicate the level of theiragreement or disagreement with the following statements on a five-point scaleranging from strongly disagree to strongly agree:

Mathematics is useful in solving everyday problems.It is important to know mathematics in order to get a good job.Most people do not use mathematics in -.heir jobs.Boys have more natural ability than girls.Boys need to know more mathematics than girls.Ill had lily choice I would not learn any more mathematics.

For the purposes of this chapel-. the two 'disagree' and two 'agree' categorieshave been combined. A high percentage of students in both ethnic and gendergroups (80 per cent) agreed that mathematics is usefill for solving everydayproblemsind that it is important to know mathematics in order to get a good job(85-9(1 per cent). The majority of Maori boys and girls (58 and 60 per cent,respectively) believed that most people use mathematics in their jobs, but thesepercentages were significantly less (p < (1.05) than those lot boys and girls of

ti I l I

Sharleen D. Forbes

Table 12.3: Students' perceptions of gender differences in mathematics (by percentage)

Boys have more ability Boys have more needin maths for maths


M F M F M F M F

Disagree 62 94 57 86 74 86 58 63

Undecided 29 5 30 13 19 9 26 27

Agree 9 1 13 1 7 5 16 10

European descent (71 and 77 per cent, respectively). A higher percentage of Maorigirls werc 'undecided' about whether most people use mathematics in their jobs

than students in any other group.For both groups of students, significantly more girls than boys disagreed that

boys possess more natural mathematics ability or greater need to know mathe-matics than girls (sec Table 12.3). Maori boys were the group most likely to agree

with these statements, while significantly more Maori girls than girls of European

descent were 'undecided' in this area. Further investigation will be necessary todetermine the extent to which the strong views held by Maori boys influence

those of Maori girls.Mathematics is a compulsory subject for the first nine years of schooling, that

is until the end of the second year in secondary school. Even in the tenth year,almost all students take mathematics (97 per cent in 1990), with a large majority

(75 per cent in 1990) continuing to study the subject the following year (Ministryof Education, 1991). Given this, it was expected that a high percentage of students

in all groups, on beginning secondary school, would state that they wouldcontinue with mathematics. Only for Maori boys was there a relatively largepercentage who had already decided they would drop the subject (13 per centcompared with 5-6 per cent in other ethnic groups). However, a high proportion

of Maori girls were 'undecided' (21 per cent compared with 10-13 per cent in

other groups).In summary then, at the beginning of secondary school. Maori girls as a group

were the most likely to be 'undecided' about the usefulness of mathematics,whether boys had more natural mathematics ability or need for mathematics, andwhether they themselves would continue to study mathematics when given thechoice. Particularly disturbing is the observation that, upon entry to secondary

school, the majority of all Maori students had already decided that they had nofuture in education beyond the secondary level.

Mathematics Performance

Performance was analysed for those students in the sample who sat the March and

the November tests. The tests administered in March ,ind November were

112 r

fathematics and New Zealand Maori Girls

Table 12.4: Gain in test performance (March-November, 1989) by gender and ethnicity

European

MeanStandardDeviation Median Mode Max Min

Males 302 3.79 4.81 4 4 7 -12Females 265 4.54 4.47 4 4 17 -9

MaoriMales 83 3.55 4 39 3 3 21 -5Females 79 2.95 4.37 3 3 14 -10

Note: Numbers do not add to 813 since only non-Maori students of European descent areincluded.

identical, and were scored out of forty points. The mean score for the entire samplein March was 15.45 (with standard deviation 7.25), while the correspondingNovember figure was 19.51 (standard deviation 8.35). In other words, this is a

mean improvement during the school year of only 4.06 marks, or 10 per cent. Thissmall gain and the accompanying large standard deviation, coupled with theobservation that one in every six students showed no improvement at all, are causefor grave concern.

The situation is worst for Maori students. Not only do they enter secondaryschool with a lower mean score than their Europan counterparts. they also exhibitsignificantly lower average gains in marks throughout the year. By November. themean difference in score between Maori student; and those of European origin was5.7 marks in favour ofthe latter. Furthermore, although there is a positive correlationbetween parents' occupations or education levels and student performance, whenthese variables are controlled for, large difkrences in test achievement continue toremain between the two groups. Thus, the mathematics performance of Maoristudents ca,iot be explained simply by referring to their low socio-economic status,and it is quite possible that 'cultural oppression' has played a role here.

As illustrated by Table 12.4, Maori girls experienced the lowest gains inachievement during the school year. However, it is also the case that, when themean scores of 'intact cLsses' were compared, one of the classes that witnessed thehighest gain relative to its March score contained only Maori students of whom60 per cent were female. This was a class of students in a small, rural, bilingualschool with a young, male Maori teacher. Although the class had the lowest meanscore in March, it gained an average of 4.6 marks over the course of the academicyear. Given this, it seems plausible that a combination of teacher enthusiasm andappropriate learning environment could make a critical difference in outcomes forMaori students in learning mathematics.

Table 12.5 provides a comparison of the November 1989 test scores with thoseobtained by students in the SIMS survey (Department of Education, 1987). Whilein 1981 the gender difference in both groups was in favour of the boys, in 1989we witness an interaction between gender and .thnicity, that is, only tbr Maoristudents is the mean score for girls lower than that of boys.

The improvement in the mean score for girls of European descent is

-: / /3

Shar leen D. Forbes

Table 12.5: November Test Performances by gender and ethnicity for the years 1989 and1981

--.. ......Mean N

---.. ......._. Standard1989 (1981) Deviation 1989 (1981)

EuropeanMale 20.7 (20.8) 8.59 304 (2076)

Female 21.1 (20.0) 8.19 267 (2037)

MaoriMale 15.7 (15.9) 6.59 86 ( 310)

Female 14.6 (14.4) 6.19 80 ( 269)

Note: Numbers do not add to 813 since only non-Mavi students of European descent areincluded.

encouraging for campaigns such as 'Girls Can Do Anything' (New Zealand'stelevision and school promotion run by the Department of Labour in 1983), aswell as for intervention programmes such as 'EQUALS' (1989) and 'Family Math'(Stenmark et al., 1986), which have been actively promoting mathematics for girls.However, these endeavours do not appear to have had the same positive impact onthe participation and performance of Maori girls. Perhaps this is not surprising,since the majority of the teachers and administrators involved in these programmesare not Maori themselves. With this in mind, it seems likely that Maoriinvolvement and ownership of the development of approaches will be necessarybefore sil:nlar improvement will be witnessed in the mathematical participationmd achievement of Maori girls.

Impact on Further Mathematics

Given that ethnic differences in mathematics performance are apparent at thebeginning, and continue to the end, of the first year of secondary school, onemight expect these difkrences to persist through the secondary years. Indeed, aswe shall see, this is the case. In the normal course of events, the students in the1989 sample would be in their third year of secondary school by 1991. It is in thisyear. Form 5, that the majority of students sit their first national examination, theSchool Certificate. Although mathematics is optional at this stage. it was thesecond most popular subject, after English, in the 1991 School Certificateexamination for both groups of students under consideration. It is not, however,surprising, given the early differences in performance and attitudes of the students,that while mathematics was taken by 84 per cent of non-Maori School Certificatecandidates who were in their third year of secondary schooling in 1991, it wastaken by only 67 per cent of the corresponding Maori candidates.

Candidates who pass the School Certificate receive a grade ranging from D(lowest) to A 1 (highest). In addition to the previously discussed differences inparticipation, there were also significant differences in achievement as shown in

1 14

Mathematics and New Zealand Maori Girls

Table 12.6: Percentage of 1991 School Certificate graduates in each grade category

Al A2 131 B2 Cl C2 DNumber ofCandidates"

MaoriFemales 0.9 4.1 11.4 21.7 33.7 25.1 3.1 2688Boys 1.5 5.3 14.1 24.1 30.9 20 9 3 2 2682Total 1.2 4 7 12 8 22.8 32 3 23 0 3.2 5370

All 5 6 12.4 21 8 23.8 22.6 12.3 1 5 39394-< --- _ -- _

Table 12.6. The most common grade for Maori students was CI . compared withB2 for non-Maori candidates (Ministry of Education, 1992).

This apparently bleak situation continued into the fourth and final year ofsecondary school. Form 6, with only 57 per cent of the Maori 1991 SchoolCertificate mathematics candidates (59 per cent of girls and 52 per cent of boys)taking the Sixth Form Certificate in mathematics one year later. This stands inmarked contrast to the 73 per cent figure for the corresponding group of non-Maori students (New Zealand Qualifications Authority, 1993).

When the small group of high-achieving (with grades of Al) Maori girls in1991 School Certificate mathematics was examined more closely in terms of theirperformance in Sixth Form Certificate mathematics the following year, these girlsscored at least as well as their female peers of European origin. The same, however,did not hold true for Maori boys (Forbes and Mako, 1993). This observation mayindicate that the few Maori girls who continue with mathematics to this level arethe very high-achieving. Despite this, throughout secondary school Maori girls.in general, exhibit the lowest participation and achievement levels in mathematicswhen compared either to Maori boys. or to New Zealand students of Europeandescent.

Conclusion

To summarize, in this chapter. I have shown that on entry to secondary school, Maorigirls are 'undecided' about the usefulness of mathematics, whether they shouldcontinue to study mathematics, and whether mathematics is a subject of greaterimportance to boys. Their average performance in mathematicsat age 13 is lowerthan that of Maori boys, in contrast with the average performance of girls ofEuropean descent which equals that of their malepeers. Furthermore, the observedlow matheniatics performance of13-year-old Maori students compared with that ofstudents of European descent increases throughout secondary schooling.

It appears that strategies to increase the participation and achievement of girlsin mathematics in New Zealand are having a positive impact on girls of Europeandescent, but are failing to meet the needs of Maori girls. The lattergenerally derivefrom low-income familiesind commence secondary school with ambivalent

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Sharleen D. Forbes

views towards mathematics. However, these variables do not account for all theobserved differences between Maori and other girls. Both the learning environ-

ment and the individual teacher may be critical factors in determining the successof Maori girls in mathematics. Maori girls belong to a culture that has suffered

from misguided attempts at assimilation by the dominant group in New Zealand

socicty, those of European descent. These attemptshave culminated in a major loss

of traditional Maori knowledge and language. It is likely that precisely targeted and

Maori-driven strategies will be needed to increase the participation, achievement,

and enjoyment in mathematics of Maori girls.

ReferenceS

El I FY, W.B. and hwING, J.C. (1985) 'The Elley-lrving Socio-Economic Index, 1981Census RtNision', N'eu, Zealand Journal of Edumtional Studies, 20,2, pp. 115-28.

DAYIES, L. and Nictiot t , K. (1993) 7i. Maori i rota i iva ?!.fahi Whakaakoranr: Maori inEducation, Wellington, Ministry of Education.

DEPARTMI NI 01 EDLR'm IoN (1987) Mathematics Achimement in New Zealand Secondary

Schools: A Report on the Conduct in New Zealand of the Serol..1 International Matlwmatia

Study within the International Association J'or the b:miltw:ion of Educational Achievement,

Wellington, Department of Education.EQUALS (1989) Assessment Alternatives in Alathematia, Lawrence Hall of Science.

University of California at Berkeley.Folushs, S.D., B1 1 I lit, T., CI ARM . M. and ROBINSON, E. (1990) Mathematics for All?,

Wellington, Ministry of Education.Foluit s, S.D. and MAKO, C. (1993) 'Assessment in Education: A Diagnostic Tool or a

Barrier to Progress', Paper presented at the 1993 World Indigenous PeoplesConference: Education, Wollongong, Australia.

Giuu N. R.A. (1984) Mathematics and Ethnicity, Wellington, Department of Educadon.

J.C. and Et Lt.y, W.B. (1977) 'A socio-economic index for the female labour force

in New Zealand', New Zealand Journal qf Educational Studies, 12.1 and 2, pp. 154-63.

Jot IN',ON, R. (1985) A Revision of Sodo-Ewnomic Indices .1Vr Neu' Zealand. New Zealand

Council for Educational Research.MINISTRY 01 EDUCA I ION (1990) Maori Education Statistics 1989, Wellington, Ministry of

Education.MINIs I We 01 EDUCAI ION (1991) Education Statistics of New Zealand 1990, Wellington,

Ministry of Education.MINIS I KY 01 El R.1( 'A I l( )sl (1992) 1991 New Zealand Senior School Awards and Examinations,

Wellington, Ministry' of Education.NI W LI Al A NI / QLIA I II ICA 1 IONS Au I I 1011.11 Y (1993) Private Communication, Wellington.

Si I NMARK, J.K., Ttiomst)N, V. and Cc )sst v, R. (1986) Family Math, Lawrence Hall of

Science. University of (:alifornia at Berkeley.T1 PUN I KOK 11(1 (1993) Paagarau Mori Mathematics and Education. Wellington, Ministry

of Maori I )evelopment.WII Y, H. (1986) 'Women in mathematics: Some gender ditThrences from the lEA

'International Association for the Evaluation of Educational Achievements' Survey in

New Zealand', New Zealand Mailwmatks Alggazilw, 23,2, pp. 29-49.\MING. K. (1988) Analysis if the Third Form Data from the 1981 WA Matlwmatics Study,

Mathematics Honours Project, Wellington, Victoria University.

116

Chapter 13

Equal or Different?: Cultural Influenceson Learning Mathematics'

Gilah Leder

Introduction

The contents of the influential Handbook of Research on Mathematics li'achim andLearnim (Grouws, 1992) illustrate clearly the wide range of thctors believed toinfluence mathenratics learning. In addition to learner and classroom-relatedvariables, reference is also made to the importance of the broad social context inwhich learning takes place. For example, the comprehensive model proposed byEcs.:es (1985) to explain students' decisions to enrol in mathematics courses (or optout of them) includes the cultural milieu, the value-system adopted by, orattributed to, important socializers, as well as students' perceptions of the socialclimate in which educational and.career choices are made.

The pervasive influence of external and societal fictors is frequently examinedwhen gender differences in mathematics learning are discussed (Leder, 1992).Listed among such factors are the influence of important others teachers,parents, members of the peer group as well as environmental and organizationalinfluences. In this chapter, (print) media references to male and female roles,achievements, and behaviours are used as a measure of societal attitudes, beliefs andexpectations, that is, of the general climate in which learning occurs and in whicheducational and career choices need to be made. The sources of interest arebroader than the ones of earlier work (Leder, 1984, 1986, 1988) in which Iconcentrated on the ways successful females were portrayed in the print media inone country and discussed these accounts from a social-psychological perspective.This study focuses on print media portrayals in two ostensibly similar countriesAustralia and Canada and discusses these media images in two ways in termsof feminist perspectives as well as the social-psychological approach (theexpectancy-value theory of motivation) used in the earlier work. Implications ofthe cultural climate described for students' perceptions about mathematics and itsrelevance to them are also explored.

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Garth Leder

Social Context

Australia and Canada aie geographically large and have comparatively smallpopulations, approximately seventeen million, and twenty-seven million inhab-itants, respectively. In both countries, the bulk of the population is concentratedin a small number of cities. Women comprise approximately 51 per cent of thepopulation and just over 40 per cent (41.8 per cent in Australia, 40.8 per cent inCanada) of the labour force. Of female employees, 55 per cent in Australia, and58 per cent of those in Canada are still concentrated in two major occupationalgroups: clerks and salespersons. Canadian statistics indicate that the proportion ofwomen in these traditionally female occupations rose from 55 per cent in 1971 to58 per cent in 1986. A comparable increase in such occupations over a similarperiod of time was reported for Australia by Sampson (1993). Data for 1991 showthat approximately 7 per cent of Australian working women are in managerial andadministrative positions. Canadian data for 1988 reveal that 10 per cent of thefemale workforce in that country are in such positions. Possible differences in jobclassification procedures make comparisons such as these problematic, however. Inboth countries, women are much more likely than men to hold part-time jobs:some 40 per cent and 25 per cent of all female employees in Australia and Canadarespectively compared with one in ten working males in both countries.'

The Print Media

The media are generally regarded as powerful reflectors and determinants ofbehaviour (Eysenck and Nias, 1978; Faludi, 1991; Jacobs and Eccles, 1985). Jacobsand Eccles (1985) reported that parents' belielS, in general and for their Ownchildren, about gender differences in mathematics learning became more ster-,typed after they were exposed to media coverage of a heated debate about themore likely reasons (environmental or genetic) for gender differences in mathe-matics performance. 'Media absorption', they argued, 'involves self-selectionaccording to previous attitude' (Ibid., 1985, p. 24).

While the impact of television is outside the scope of this chapter, it is

nevertheless of interest to note that the long-running series, Sesame Street, oftencited as an effective enrichment programme for socially disadvantaged children,'has achieved a special status as Part of the cultural milieu from which children arethought to acquire a wide range of early childhood competencies early numbercompetencies among them' (Secada, 1992, p. 641).

There is often concern about the manner in which issues are reported by themedia. Faludi (1991) stressed the implications of biased reporting on women'slives:

The Victorian era gave rise to mass media and mass marketing twoinstitutions that have since proved more t a:a-nye devices for constrainingwoinen's aspirations than coercive laws and punishments. They rule with

118

Equal or Different?: Cultural Ityinences on Learning Mathematics

the club of conformity, not censure, and claim to speak for female publicopinion, not powerful male interests. (Faludi, 1991, p. 48)

She further argued that incalculable harm was being done by contemporary mediastories:

The press was the first to set forth and solve for a mainstream audience theparadox in women's lives . .. women have achieved so much yet feel sodissatisfied; it must be feminism's achievements, not society's resistance tothese partial achievements, that is causing women all this pain. (Ibid.,p. 77, my emphasis)

That journalists themselves are aware ofthe limitations oftheir own reporting orthat of their colleagues is made explicit from time to time. In an article published in aCanadian newspaper, the Globe and Mail, headed in part 'on the sins and omissions ofthe anglo press', one columnist described as biases ofthe English-Canadian media intheir coverage ofQuébec and constitutional issues: blatant distortions, gross mistakesand flawed analysis or knee-jerk stereotyping (Gagnon, 1992, p. D3).

A perceived need to examine how issues pertaining to gender equity, towomen's roles and lifestyles are portrayed in the popular press in two apparentlysimilar countries. Australia and Canada, served as the stimulus for this contribu-tion. Such a review should reveal the broad cultural climate in which students learnand in which they need to decide among other issues whether or not tostrive for success in mathematics and whether to continue with mathematics oncethe subject is no longer compulsory. Such decisions have important implicationsfor long- term career and life opportunities. The ways in which mathematicsqualifications act as a critical filter to tertiary courses and apprenticeships have beenwell documented (Sells, 1973; Tobias, 1993).

The Study

Two newspapers with a large circulation and readership were monitored over atwo-week period in April and May of 1992: the Globe and Mail in Canada and theAge in Australia. Articles about gender-equity issues, about women's roles, andones that contrasted female and male lifestyles were of particular interest. TheGlobe and Mail was monitored independently by a second reader. Interrateragreement was 94 per cent. Only articles selected by both readers are included inthis report.

Once selected, a content analysis was made of each of the articles using twodistinct methods.

The Social-Psychological Approach

The expectancy-value theory of achievement motivation has been widely used toexplain and predict behaviour (McClelland et al., 1953). Within this framework,

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Gi lab Leder

it is argued that individuals are more likely to strive for goals which they value andbelieve they can attain. A description of an occupation as male-dominated and'uncomfortable' or difficult for females affects both the value and perceivedlikelihood of attaining that ocCupation. Educational paths selected, or advised, (forexample, participation in mathematics and related careers) are likely to be shapedby these perceptions (Eccles, 1985).

Based on previous media searches, a number of common themes wereanticipated: that females are disadvantaged though the fields in whichdisadvantage might occur were not predicted; that male and female roles continueto be different; and fear of success imagery. Relevant to this last category wereoccurrences where success was described as being attained through luck orwithout effort, instances where success in a career was said to have been achievedat a personal cost, confirmations that it was more difficult for females than malesto be successful, and gratuitous references to interpersonal relationships orsignificant others.

Fenthtist Perspcaines

Three major themes were anticipated, conveniently designated as liberal femiii-isms. social feminisms, and feminisms of difference (compare with Mura, Chapter19, this vOlume). Given recent explorations of links between feminist pedagogyand mathematics (for example, Damarin, in press; Jacobs, 1994), an examinationof print-media publications for feminist themes seemed timely.3 Jacobs (1994), forexample, argued that 'aspects of feminist pedagogy can be used to make womenmore willing to study mathematics, and therefore learn mathematics' (Jacobs,

1994, p. 12).Articles which included explanations of gender imbalance in terms of barriers

that restricted women's access to current social benefits and particular sectors ofthe labour market were deemed to be consistent with liberal feminisms if theircontent implied that the imbalance could be eliminated by the removal of barriersand the re-socialization of women. In such articles it was often argued that femalesshould be given equal opportunities to access the elements of the current socialsystem which are defined within that system as positive and desirable.

Articles which pointed to a gender imbalance and explained this as a result ofthe contemporary social structures, rather than women's failure to fit into them.were considered consistent with social feminisms. Here the current social structurewas regarded as a mechanism of domination, oppression and exploitation offemaks so that the structure itself would need to be changed before women aswomen would be able to participate successfully and willingly in all spheres.

Items with a theme that gender imbalance was the result, in our culture, ofthe centrality of masculine values and could he redressed only if the positivedimensions of women's nature were identified and promoted, were considered asrepresentative of feminisms of difference. In this category experiences, desires andspirituality of females would be regarded as essentially different from those ofmales.

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Equal or Different?: Cultural Influences on Learning Mathematics

The three broad feminist perspectives outlined above are not mutuallyexclusive. There is, in particular, an overlap between social feminisms andfeminisms of difference because both understand women to have different needsand experiences from those of men. An important difference between these twoperspectives is the social feminists' view that the reform of society required toinclude female needs and desires would be for the good of both women and men,and that feminine qualities are not necessarily women's qualities, that is, genderdifferences have a social origin. Feminisms of difference, on the other hand, areseparatist in that they understand these qualities to be essentially female and,therefore, that females need different social structures, not only from the currentones, but from the ones of males. Both social feminisms and feminisms ofdifference criticize the liberal approach because it identifies women and femininityas the problem and implies that women need to be given the opportunity to bemore like men if they are to succeed.

The argument that females' experiences and subjectivity are different fromthose of males is common to all feminisms. It is the perceived origin of thisdifference that distinguishes between them: exclusion of women from legitimizedrealms of society for liberal feminisms; the oppressive and exploitative nature oflegitimized society and of the laws that legitimize that structure for socialfeminisms; and the need to recognize, accept, value, and legitimize essentialdifferences for feminisms of difference. All three perspectives point to directionsin which mathematics might be reconceptualized and new strategies be introducedinto the mathematics classroom (Damarin, in press; Jacobs, 1994; Mura, Chapter19, this volume).

Results

Articles dealing with gender-equity issues or highlighting successful womenappeared in each newspaper for each day the papers were monitored. Their lengthand frequency varied. Some typical examples are shown below.

Example 1Headline: 'A prima donna from the start'Summary: A Swedish soprano had an incredible debut and became a star

overnight. 'It was pure luck', she recalls. Her Austrian husbandis a baritone. Her sons are now old enough for her not to worrywhen she goes on tour. (Setting: success in music)

Coding: 1. Fear of success imagery: success was 'pure luck', gratuitousreference to others.2. No evidence of feminisms theme.

Example 2Headline: 'Getting mixed messages on being a woman'Summary: A series of articles which convey conflicting messages about the

121

Gilah Leder

role of contemporary women is summarized. 'Every workingwoman with children knows what it means to be caughtbetween conflicting hopes for herself and the expectations offamily."Even if such things as good day care, understandingemployers and supportive husbands were universally guaranteedto working mothers, I'm not sure that women could ever fullyavoid the "discontinuity" of a mother's life."Men are still callingfor the shots.'

Coding: 1. Fear of success imagery: conflict between career andmotherhood needs.2. i) Social feminisms: evidence of imbalance in contempo-

rary social structures;ii) Feminisms of difference: reference to difficulty of'equalizing the roles of men and women', given thecentrality of masculine values.

Example 3Headline: 'Plans for tougher affirmative action'Summary: Tougher penalties should be imposed against employers who

discriminate against women in the workplace. Smaller com-panies should no longer be exempt from the Affirmative ActionAct. Many companies have 'done very little to improve oppor-tunities for women'.

Coding: Liberal feminisms: gender equity (deemed to be desirable) hasnot been achieved in the workplace but better legislation shouldlead to improvements.

Example 4Headline: 'Former dancer keeps House in order'Summary: Miss Boothroyd 'made history' when she became the first

woman to be elected Speaker of Britain's House of Commons.As one of sixty women in the 651 member House she hadearned a reputation for fairness and impartiality as the DeputySpeaker. She is 'a former chorus-line dancer and secretary'.(Setting: success in politics)

Coding: 1. Fear of success imagery: 'It will be a lonelier life than I haveever known before.'2. Liberal feminisms: Previous experience (her record asDeputy Speaker) indicates that she is able to overcome real and/or perceived barriers and to compete successfully in a tradition-ally male domain. Her performance has been, and is likely tocontinue to be, judged by male standards.

Example 5Headline: 'Women coming to grips with math ghost'Summary: Like other students, female student teachers often find mathe-

matics more difficult than their male peers. Many universities

122

Equal or Different?: Cultural lqiuenas on Learning Mathematks

have now introduced courses 'explicitly aimed at eiadicatingmath phobia'. Schools are also more aware that girls need to beencouraged in mathematics classes.

Coding: 1. Perpetuation of stereotype: Females often have more diffi-culty than males with mathematics.2. Liberal feminisms: Females should aim to perform equallywith males. With appropriate assistance, there need be nodifferences in the performance of females and males (in mathe-matics).

Discussion

Both newspapers contained articles with fear of success imagery and feministthemes. A number of these focused on the fate of females generally. In others themetaphors were introduced in stories about specific individuals. The lack ofprogress made towards gender equity was discussed in several articles.

Both newspapers contained articles which confirmed that females werewidely disadvantaged in politics, in the arts, in the workforce generally, in jail,in the lifestyle thrust upon them. Several of these stressed differences betweengroups while ignoring differences within groups. Interesting' y, males weredescribed as being disadvantaged in one (Canadian) article, females as beingspecifically advantaged in two (Australian) articles. One of the latter describedfemales' higher pass rates at senior high-school level in a wide range of subjects.

In both newspapers there were descriptions of affirmative action practices,with 'Women fighting back' the dominant theme in two of the (Canadian)articles.

Comparisons between gender and race discrimination were made in severalof the (Canadian) articles. During the two-week period monitored, this morecomplex issue did not attract the attention of the Australian media.

The daily appearance of articles which confirmed traditional stereotypes,dwelt on the difficulties faced by females, stressed familycareer conflictsexperienced by them, focused on societal pressures and/or the prevalentacceptance of male standards and values is noteworthy. While the frequency withwhich they appeared, as well as their placement in the paper (rarely on the frontpage) should be balanced against the large number of other issues discussed eachday, their impact should not be underestimated. A further, intensive testing ofJacobs and Eccles' (1985) conclusion that 'media absorption involves self-selectionaccording to previous attitude' (p. 24) is clearly warranted. Identifying, discussing,and exploring the origins of the areas of disadvantage depicted (for example,unequal opportunities for participation in selected areas, social or organizationalbarriers, the [in]appropriateness of using male behaviours or standards as the norm)may serve to counteract negative images and point to constructive initiatives forthe mathematics classroom.

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Gilah Leder

Concluding Comments

Gender differences in mathematics learning have attracted much research atten-tion. Reference is frequently made in this work to the influence of the socialclimate and environmental factors. An examination of the contents of two largeand widely read metropolitan daily newspapers published in two different, highlyindustrialized countries, revealed the regular inclusion of articles which morefrequendy confirmed, than challenged, established male values and powerstructures, and dwelt on traditional stereotypes and the difficulties faced by femalesin particular in juggling professional and personal pressures. The impact of thesemessages on adolescents and their parents requires careful monitoring. Meanwhile,it appears that the subtle messages conveyed in the popular press are consistent withsmall but consistent differences in the ways females and males perceive and valuemathematics and related careers as appropriate for themselves.

Notes

An earlier version of this chapter was presented at the 10WME Study Group held aspart of the ICME-7 in Québec City, Canada, August 1992.Sources: li;mten at It ink (1991) 13. 3. Australia, pp. 12-13 and 1992 Canada Year Book.Ottawa, Statistics Canada.

3 The contribution ofJulianne Lynch to this analysis is gratefully acknowledged.

References

DAMAlk S.K. (in press) 'Gender and mathematics from a feminist standpoint'. in Si ("AIM.W.G.. MA. E. and Byisii. L. (Eds) Neu. Directions in Equity .for MathematksEdmation. Cambridge. Cambridge University Press.

Ec II s, J.S. (1985) 'Model of students' mathematics enrolment decisions', EducationalStudies in Mathemarks. 16, 3, pp. 311-14.

Eysl !';(K, H.J. and NiAs. D.K. (1978) Sex, Violence and the Media, New York, Harper andRow.

FAI ui )1. S. (1991) Backlash. New York, Crown.GAI,N( IN. L. (1992) 'On the sins and omissions of the anglo press'. in DAI J. and L.

Pu Ms! L. Globe and Mail. May 9, Section 1)3.Cut; wws, D.A. (Ed) (1992) Handbook of Research on Alathimatics li.aching and Learnnw. New

York, MacMillan.JA«uss. J.E. (1994) 'Feminist pedagogy and mathemducs'. Zentralblatt tic, Didaktik der

Mathematik, 26, 1, pp. 12-17.J.E. and E«.1 s, J.S. (1985) 'Gender difference% in math ability: The impact of

media reports on parents', Educational Researcher, 14. 3, pp. 20-4.1.11u is, GA: (1984) 'What price success? The view from the media'. The Exceptional Child,

31, pp. 223-4.Lt1)1 Is, G.C. (1986) 'Successful females: Print media profiles and their implications'. The

.1ournal (4. l'sychology. 120, pp. 239-48.

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Equal or Different?: Cultural Ittfluences on Learning Mathematics

LEDER, G.C. (1988) 'Fear of success imagery in the print media', The Journal of Psychology.

122. pp. 305-6.LEDER, G.C. (1992) 'Mathematics and gender: Changing perspectives', in GROLWS, D.A.

(Ed) Handbook of-Research on Mathematics Waching and Learning, New York, MacMillan,pp. 597-622.

McCIEI LAND, D., ATKINSON, J.W., Ca ARKE, R.A. and Lowia r. E.L. (1953) The

Achievement Motive. New York, Appleton. Century, Crofts.NEw Sou in WALES ANTI-DISCRIMINATION BOARD (1984) Discrimination and Reltgious

Conviction, Sydney, NSWADB.SAMPSON, S.N. (1993) Dismantling the Divide, Canberra. Australia. Department oc

Employment. Education and Training.SECAI Wk. W.G. (1992) 'Race, ethnicity, social class, language and achievement in

mathematics', in Giwews. D.A. (Ed) Handbook of Research on Mathematics Teaching and

Learning. New York, MacMillan. pp. 623-60.s, L. (1973) High School Mathematics as the Critical Filter in the Job Market (ERIC No.ED080 351), Berkeley, University of California.

TotuAs, S. (1993) Overcomiv Math Anxiety. revised edition, New, York, W.W. Norton andCo.

125

Chapter 14

Gender Inequity in Education: ANon-Western Perspective

Saleha Naghnii Habibullah

In this chapter. I challenge the popular notion that western strategies forovercoming gender inequity in education are likely to be equally effective in non-western societies. In my opinion, there is a direct link between a country's socio-cultural conditions and the manner in which its people perceive and react to theissue of gender imbalance. For this reason, there exist enormous differencesbetween the western world and many of the developing countries regarding thisissue. Consequently, for many years to come, short-term goals and strategies foreducational improvement will have to , ary across countries in accordance witheconomic conditions and socio-cultural realities. In this regard. I suggest that abroad-based programme of research may prove to be vet y beneficial.

The issue of gender inequity in mathematics education must be considered inconjunction with the broader issue of gender imbalance in education as a whole.In this regard, it is important to understand thc differences between theeducational contexts in the underdeveloped and the developed worlds. From apurely theoretical viewpoint, one of the primary goals of education is to foster thedevelopment of the human inind so that individuals are able to make appropriatechoices, succeed in professional and familial spheres, and indulge in a variety ofcreative activities in accordance with their inherent potentialities. While there islittle doubt of this being an ideal that every country sliould aspire to in the longrun, the sheer heterogeneity of today's world requires flexibility, thoughtfulnessand careful consideration of what is and what is not feasible in a particular culture.Although we may overlook differences in the history. culture, and socio-economicconditions within each of the developed and underdeveloped realms, thedisparities that exist between them cannot be ignored. Numerous facilities that aretaken for granted in affluent societies may be scarce or absent in poorer nations.Furthermore, social norms considered appropriate in many non-western countriesmay be deemed completely unacceptable in western culture ano ice versa. Takingthese observations into account, it is clear to me that short-term educational goalsmust vary across countries in accordance with local cultural factors and socio-economic conditions.

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The particular characteristics of each country, such as the socio-economic andcultural factors alluded to above, have a direct bearing on the manner in which itspeople perceive and react to the issue of gender inequity. As such, gender inequityin education does not present similar concerns in all parts of the world. A numberof arguments can be employed to elucidate this observation. First, in countrieswhere the procurement of basic necessities is the primary concern, gender-equitable education has a much lower priority. In such countries, for possiblymany years to come, it may be only fair to focus the limited resources on povertyreduction, rather than on gender equity in education. Similarly, and this brings meto my second point, in countries with high illiteracy rates, and in some casesconsiderably higher illiteracy rates among women than among men, it is probablyonly fair to make reducing illiteracy a matter of top priority Only after this hasbeen achieved, c n resources be devoted to more subtle issues such as reform Jfthe male-oriented primary and secondary-school mathematics curriculum.

Third, in countries where, for religious or historical reasons, the male adultis accepted as the bread-winner for the family, pursuit of employment is a muchgreater concern for men than for women. Given the link between advancededucation and higher paid jobs, it is not surprising that, in such countries, boys aremore likely to aspire towards higher education. In such societies, efforts to reducegender bias at the primary and secondary levels of education may prove to be moresuccessful than attempts to reduce gender imbalance in higher education.

Finally, in societies where, for religious or cultural reasons, boys and girlsreceive their education at separate schools and colleges, the manifestation ofgender inequality might be quite different from that in coeducational situations.At the secondary-school level, girls in sex-segregated classes probably enjoy agreater degree of self-confidence and freedom of expression than those incoeducational environments who might feel intimidated by the presence of boyswho are generally more assertive. At the post-secondary level, however, lack ofconmiunication with male counterparts may reduce girls' exposure to situationsoutside of the domestic world and, as a result, hinder their intellectual develop-ment.

The unifying thread of the above discussion is the fact that historical, religious,socio-economic, and cultural realities all combine to shape a country's educationalsystem as well as the nmde of behaviour, the value-system, and the aspirations ofits people. As previously mentioned, the phenomenon of gender inequality ineducation does not manifest itself in an identical fashion around the globe, andthen- is enormous diversity in how different peoples perceive and react to the issue.I have attempted to show that it would be impossible to develop a uniform strategyfor educational improvement in all parts of the world. While the equitabledevelopment of humanity should remain the long-term universal goal ofeducation, short-terni educational objectives for a particular region of the worldwill necessarily have to be in i-ccordance with local needs.

To this point, the western movement for gender equity in education has notgiven adequate recognition to the aforementioned realities. In order to develop abroader, less westernized view regarding the issue of gender imbalance, to widen

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eople's understanding of cross-cultural differences, and to formulate appropriatestrategies for educational improvement in different pat zs of the world, it is of theutmost importance that we explore and study in detail the socio-economic andcultural conditions of various countries. Such studies will necessarily have to focus,not only on economic conditions and educational and employment structures, butalso on socio-cultural norms including people's beliefs, value-systems andaspirations. Comparative studies may prove to be extremely useful in ascertainingdifferences and similarities between nations, in determining the benefits anddrawbacks of a variety of educational systems, and in the formulation of strategiesfor educational improvement in various parts of the world. In order to accomplishthis objective, scholars from around the world should collaborate in the creationof a comprehensive, broad-based programme of research.

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Chapter 15

Gender and Mathematics: TheSingapore Perspective

Berindedeet Kaur

Introduction

Over the last two decadest major focus of educational research has been theinvestigation of sex-related differences in learning mathematics. Gender differ-ences in mathematics education have been shown to be complex and influencedby the interaction of a variety of social factors, and hence to vary considerablyacross cultures (Ethington, 1990; Hanna, Kiindiger and Larouche, 1990). In thecase of Singapore. only a handful of studies to date (Chung, 1985; Kaur, 1987;Leuar, 1985; Ministry of Education, 1988; Tan, 1990; Tanzer and Sim, 1991) havecontributed to the growing body of literature in this area. This chapter presentsthe Singapore perspective on gender issues in mathematics education through areview of the available Singapore research and discussion of their findings.

Review of the Literature

Two separate studies by Chung and Leuar, both in 1985, examined the relationshipbetween gender and mathematical ability. Each study used a sample of first-yearjunior college (age 17 years) students selected on the basis of their mathematicsgrades in the GCE 0-level (General Certificate in Education Ordinary Level)ex,mination, a national examination written by all students at the end of theirtenth year of formal schooling (age 16 years).

Chim (1985)

In her study. Chung examined the differences in attitudes and expectations ofeighty students, of whom fifty had 0-level mathematics scores of 70 per cent ormore (the 'high' group), and thirty had scores between 50 and 59 per cent (the'low' group), with equal numbers of males and females in both groups. Theassessment instrument employed was a questionnaire comprising thirty-eight items

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adapted from the International Project for the Evaluation of EducationalAchievement (Husén, 1967) and the Sandman (1979) Mathematics AttitudeInventory.

All respondents in the study viewed mathematics as an important subject,useful in the as well in soL iety at large. While individuals in the 'high'group considered mathematics to be more important than those in the 'low' group,there were no significant gender differences. Males were found to be moreconfident than females of their ability to learn mathematics, with subjects in the'high' group demonstrating a greater degree of confidence than their low'-abilitycounterparts. In terms of enjoyment, again there were no significant genderdifferences, but students in the 'high' group enjoyed mathematics more than thosein the 'low' group. Only male members of the 'high' group believed in thecommon stereotype of mathematics being a male domain and, other than the factthat males in both groups preferred trial and error methods of problem-solving,no significant gender differences were found in students' views of the nature ofmathematics. Some differences were found in students' career aspirations however.Although more females than males indicated an interest in accounting, a majorityof male students indicated an interest in studying engineering at universityindteaching appeared to be the predominant career preference for females.

Lcuar (1985)

Leuar's investigation was confined to 163 students (ninety-one boys, seventy-twogirls), all of whom had obtained good results in 0-level mathematics and werecurrently studying two advanced mathematics courses. Again, a questionnaire wasemployed. The issues investigated included: the extent to which students likemathematics compared with other advanced subjects studied; students' opinions asto whether girls or boys perform better in Mathematics, scienceind Englishlanguage comprehension and communication; students' views on the usefulnessand practicality of mathematics; and, whether students required the assistance ofa private mathematics tutor.

Data from the study indicated thatilthough their levels of achievement in0-level mathematics were similar, males significantly outperformed females on themathematics test taken at the end of the first year of junior college. Both sexesdisplayed similar patterns of affinity towards mathematics and both believed thatmales would do better in mathematics, although the boys surmised that girls wouldfare better in English language comprehension and communication. Regardless ofsexi majority of the subjects believed in the practical utility of mathematics, butdid not know whether mathematics would be useful to them in their futurestudies. Very few of the study's participants had a private tutor, and most workedindependently with occasional help from friends and teachers.

Gender and Mathematks: The Siumore Perspeaive

Kaur (1987)

In her study of gender differences in mathematics attainment, Kaur analysedperformance on specific topics in the 1986 Singapore-Cambridge 0-levelexamination in mathematics. The examination consisted of two papers, each ofwhich assessed students' understanding of basic mathematical concepts andapplications, as well as their ability to demonstrate this clearly in writing. Theexamination questions were classified by topic, level of cognitive complexity, andthe degree of spatial ability required, and the points earned for each part of aquestion on both papers were recorded for use in data analysis. From a populationof 42 627 examination candidates, with girls slightly outnumbering boys, arandom sample of 176 was drawn with equal representation of the sexes from eachof six achievement levels.

Overall, boys significantly outperformed girls on the compulsory questions.More specifically, boys did significantly better than girls on the followingcompulsory topics: mensuration, statistics, arithmetic, geometry, and probability.Girls, on the other hand, excelled in the areas of algebra and graphing. While boyssurpassed girls on the questions testing spatial ability and cognitive complexity. nosignificant gender differences were found in the areas ofcomputation andproblem-solving. On optional questions, girls preferred problems requiringalgebra, graphs, and 2- dimensional vectors, while boys' only marked preferencewas mensuration.

Ministry ty. Education (1988)

The Ministry of Education's studyis reported in Yip and Sim (1990), focused onthe relationship between examination performance, gender, and birth-month forprimary students who sat the PSLE (Primary School Leaving Examination) duringthe period 1977 to 1987. The findings indicated that at the age of 9 years, girlsoutpertbrmed boys in their first language, usually English, their second language,usually the mother-tongueind in mathematics. Students born earlier in the yearalso performed better than those born later the same year. By the time they sat thePSLE at 12 years of age, girls had maintained their superiority over the boys in thetwo languages, while the boys outperformed the girls in mathematics and science.Once again, those born earlier in the year seem to enjoy an advantage in terms ofenhanced performance.

Thn (1990)

The focus of Tan*s study was on mathematics achievement, mathematics anxiety,and locus of' control. More specifically he focused on these areas in the contextof differences between males and females, and between arts and science students.The participants in the study comprised 558 16-year-old students chosen from sixsecondary schools, with approximately equal numbers of students in each of thefour gender-stream groups. The instruments used in the study were the Fennema-

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Sherman Mathematics Anxiety Scale, and a Mathematics Achievement Locus ofControl Scale and Mathematics Achievement Test, both developed by theresearcher himself specifically for the study.

A stepwise multiple regression analysis was performed with gender, stream,mathematics anxiety, and locus of control as independent variables, and mathe-matics achievement as the dependent variable. The results of this analysis showedstream to be the best regressor, followed by locus of control and, thereafter,mathematics anxiety. Gender was not a significant regressor, implying thatmathematics achievement was not linked to the sex of the subject. Furthermore,a two-way analysis of variance showed no significant interactions between genderand stream for each of the dependent variables, mathematics achievement,mathematics anxiety, and locus of control. There were, however, significantdifferences in all three dependent variables between arts and science students, withscience students outperforming arts students in mathematics, exhibiting lessmathematics anxiety and being more likely than arts students to attribute controlto internal factors. Indeed, the correlation between locus of control andmathematics achievement was observed to be Agnificantly stronger in the case ofarts students.

liwz-er and Sim (1991)

Tanzer and Sim's study exannned self-concept and achievement attribution ofprimary-school students. Eight schools were chosen at random to participate inthe study, with the proviso that the sample be representative of the different typesof institutions existing in Singapore. From these schools, a sample of 1145 studentswas drawn, with 41 per cent aged 10 years and the remaining 59 per cent aged 12years, and roughly equal numbers of boys and girls. The students represented adiversity of socio-economic backgrounds and their ethnic composition was areplication of the actual population, that is, 82 per cent Chinese, 12 per cent Mal.5 per cent Indian and 1 per cent from other ethnic groups. A battery of scalesmeasuring self-concept, achievement attribution and test anxiety were admin-istered to the students in their respective classes by their teacher.

Gender differences in self-concept and attributional style were found to bedomain-specific and consistent with prevailing stereotypes. In the younger agegroup (10-year-olds), boys' self-concept in regards to mathematics and school ingeneral was higher than that of the girls, but this difference decreased with age. Inaddition, the younger boys were more likely to attribute their success inmathematics to eftbrt than the girls were. Older students (12-year-olds), mole thanyounger students, and irrespective of their gender. attributed their success inreading and in mathematics to external factors, such as luck and ease of task. Girlsmore readily than boys believed that failure in mathematics was the result of lowability, and failure in reading and ni mathematics were attributed to poor efibrtmore by the older students than by the younger ones. In addition, the older pupilswere likely to view such failures as more the result of external than internalcircumstances.

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Gender and Mathematics: The Singapore Perspective

Discussion

Examinations play a vital role in Singapore's educational system (Yip and Sim,1990, p. 147). The studies conducted by both Kaur (1987) and the Ministry ofEducation (1988; Yip and Sim, 1990) utilized available national examinations, andtherefore, were able to involve the entire cohort of students at the desired levels.In terms of overall achievement in mathematics, the findings of both studiesconcur with the results in the wider body of related literature. However, thefinding of no significant gender differences in the area of computation andproblem-solving (Kaur. 1987) is in conflict with research findings from othercultures (see, for example, Sabers, Cushing and Sabers, 1987; Martin and Hoover,1987). A potential explanation for this may lie in the fact that the preparation ofstudents for the GCE 0-level examination involves a substantial amount of timebeing spent practising previous examination questions and that the problem-solving questions included in the examination are quite routine.

What cannot, however, be achieved through a process of drill and practice isconceptual understanding of mathematics. At the advanced level of study inmathematics, where an in-depth understanding of the subject is vital, boys werefound to outperform girls (Leuar, 1985). Improving girls' conceptual under-standing of mathematics may, indeed, be the key to improving their mathematicsachievement at the secondary level.

In contrast with Kaur's (1987) findings, Tan (1990), as previously mentioned,found no significant gender differences in mathematics achievement. mathematicsanxiety, and locus of control among 16-year-olds. He suggests that the tenuousrelationship between affective variables and achievement may be the result of therapid modernization of Singapore societ y. and consequent emancipation ofwomen. However, since his sample was not designed to be representative of16-year-old Singapore students. generalizations should be treated with caution.

At present, there is no comprehensive study of Singapore students whichmight indicate the direct and indirect influences on achievement levels andorientations towards the learning of mathematics. In describing models ofachievement behaviour, Ethington (1992) ascribes different paths to males andfemales. Support for this model is found in three of the studies reviewed here(Chung, 1985; Leuar. 1985; and Tanzer and Sim, 1991), which suggest that theremay be ditkrent paths to understanding for male and female Singapore students.

Conclusion

At 16 years of age, boys outperform girls at 0-level matheniatics, a compulsorysubject in Singapore schools. Given that a pass in mathematics is essential for entryto most tertiary institutions, why then do females appear to be lagging behind%Singapore is a traditionally male-dominated society in which mathematics andrelated skills are regarded as strictly male capabilities. Malathy (1992) argues that,as a result, girls tend to shy away from this 'masculine pursuit' and she suggests that

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the key to change may lie in the reform of teaching. In a fairly recentcomprehensive review on mathematics education in Singapore, Khoo et al. (1991,p. 29) state that 'many questions remain to be explored in the area of gender andmathematics attainment, least of all, the true extent of any difference and thefactors that may possibly account for the difference'. It is clear that morecomprehensive, Singapore-based research will be necessary in order to assess theextent of any gender-related differences in mathematics education and developstrategies for redressing any observed imbalance.

References

CHUNG. YY. (1985) 'Investigation of Sex and Ability-Based Differences in Attitudes andExpectations in Mathematics', Unpublished assignment during the period of induc-tion, Singapore Institute of Education.

E ii um; i m. C.A. (1990) 'Gender differences in mathematics: An international per-spective' Journal jOr Research in Matlumatics Edwation, 21,1, pp. 74-80.

E riliNG1()N. C.A. (1992) 'Gender differences in a psychological model of mathematicsachievement'Journalfor Research in Mathematics Edwation, 23,2, pp. 166-81.

HANNA, G.. KLINDIGFR, E. and LAWL1(71 if., C. (1990) 'Mathematical achievement of grade12 girls in fifteen countries', in BOK I ON, L. (Ed) Gender and Mathematics An

International Perspective, London, Cassell, pp. 87-97.HlA.N, T. (Ed) (1967) International Study of Achievement in Mathematics A Comparison of

7ivelve Countries, New York, Wiley.KALA, B. (1987) 'Sex Differences in Mathematics Attainment of Singapore Pupils',

Unpublished masters dissertation, United Kingdom. University of Nottingham.io(), P.S., T.H., FOONG, PY., KAUR, B. and LINI-TEo, S.K. (1991) A State-of-the-

Art Review on Madwmatics Education in Sivapore, Singapore, Institute of Education.LI.UAR, B.C. (1985) 'Sex Difkrences and Mathematical Ability% Unpublished assignment

during the period of induction. Singapore. Institute of Education.MAI Art iv. K. (1992) 'Math Must gas do worse than boysF, Young Parents, March-June,

Singapore, Times Periodicals, pp. 32-7.MARFIN, D.J. and Hooyl R, H.D. (1987) 'Sex differences in educational achievement: A

longitudinal study*.fournal of Early Adolescence, 7 , 1, pp. 65-83.Mixi'i kv OP EDI:CAI ION (1988) Analysis of Performance by Sex on the PSLE 1977-1987,

Singapore. Ministry of Education.SAW ItS, D., Cum IING, K. and SAIii.nN D. (1987) 'Sex differences in reading and

mathenutics achievement for middle school students'. Journal of Early Adolescen«., 7,1. pp. 117-28.

SAM )MAN, R.S. (1979) Madwmatics Attitude Inventory Minnesota Research and EvaluationCmter, University of Minnesota.

FAN. as. (199(t) 'Mathematics Anxiety, Locus of Control and Mathematics Achievementof Secondary School Students'. Unpublished masters dissertation, Singapore. NationalUniversity of Singapore.

N.K. and Sim, Q.E (1991) Self-Conwpt and Athin'ement ,guributions: A Study ofSinoporean Primary School Students (Research Report no: 1991/5), Insfitut fiir

Psychologie der Karl-Frazens, Austria, Universitat Graz.Ynt S.K.J. and Sim, W.K. (199(t) Iii,olution of Educational Lxcellence 25 ar. of Edwation in

the Republic of Singapore, Singapore. Longm,ni

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Chapter 16

Gender and Mathematics Education inPapua New Guinea

Nee la Sukthankar

Introduction

At 14 years of age. Jenny is the oldest of six children. She serves as a secondarymother figure in her home, performing household duties such as cooking.cleaningind looking after her younger siblings. She barely has sufficient time togo to school, let alone study enough to make attendance worthwhile. While thisscenario may sound extreme, it is in no way atypical in a country like Papua NewGuinea where the average fannly has five or six children. As in any other country,culture, social beliefs, and traditions all play an important role in the education offemales. But, situations like the one described above serve only to widen the riftbetween male and female students. The aim of this chapter is to investigate thecultural issues affecting the education of females in Papua New Guinea and tosuggest ways of improving the quality of their mathematical education.

Female Enrolment in the Education System

Papua New Guinea is a country comprised of countless, small, isolated commu-nities with 700 dialects and numerous counting systems (Lean, 1987). Most peoplespeak either Pidgin or Motu in addition to their village language. According toNational Statistics Bureau data for 1988, 86 per cent of the population lives in ruralareas where good schooling is hard to find. Children are forced to walk longdistances in order to attend school and parents are reluctant to send their daughtersunder these conditions out of concern for their safety. While international schoolswith good educational facilities exist in the cities, they are far too expensive for theaverage salaried Papua New Guinean citizen to afford.

There is a substantial difference in the number of male and female studentsenrolled at the primary and secondary-school levels, with males outnumberingfemales. The gap increases at the tertiary level (Guthrie and Smith, 1980). Thissituation is in marked contrast to that in many developed countries where primary

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and secondary education is paid for through taxation and hence is mandatory.While having to pay to educate one's children does not create problems for theaffluent, families with limited means are forced to make choices. In thesecircumstances, research has shown that families choose to educate their sons beforetheir daughters. This is the situation in Papua New Guinea. In short, the issue canbe linked to the level of education which the parents themselves have received. Infamilies in which both parents were educated, it has been found that daughtersenjoy better opportunities for schooling. Female students at a variety of scholasticlevels in Papua New Guinea, including those participating in distance-educationprogrammes, were found to have originated from backgrounds in which theparents were educated (Oliver, 1985).

Social and Cultural Influences

The social differences between men and women are reflected in the choices theymake at school and university. Certain subjects such as woodwork, metal work,and the sciences are favoured by men, while others such as home economics,textiles, and child care are designated for women. Moreover, female students areexpected to be docile and retiring in an atmosphere which generally equatesboldness and flair with success. The ramifications of this are particularly serious inmathematics and science. Often the contradictions of being a women in scienceor mathematics are so overwhelming that females opt out in favour of arts-relatedsubjects. Furthermore, since the learning potential of women is not valued, it isnot encouraged. Unfortunately this attitude is not confined to families, but is alsoinstitutionalized in girls' organizations such as the girl guides and brownies. Inessence, these groups undermine the role of women in the workplace, by awardingbadges for demonstrating skill mainly in such stereotypical female-roles as'homemaker', 'childminder', and 'home nurse'. In this way, Papua New Guineansociety places limitations on the spheres of activity in which a woman is permittedto develop a competitive spirit. These are a few of the ways in which social normsconvey a different set of expectations for males and females, creating differentlifestyles for each sex, with women viewed as subordinate to men. The good newsis that there are signals that the situation is slowly improving in the cities where,as a result, the sex-related differences are gradually diminishing as well.

In Papua New Guinea society, a multitude of social and cultural customscombine to influence gender differences in education. At a broader level, it is alsopossible to identify certain customs that communicate information about sex-rolesand sex-appropriate behaviour. One of these is the established system of 'brideprice'. According to this practice, one which serves only to object4 women, awife is 'purchased' from her parents by the bridegroom's family. Since a women'slevel of education is not valued it is not reflected in her 'price'. This custom is sopervasive that, even among well-educated families, the male child is provided witha good quality international education, while the female child is relegated to aninferior community school which continues to propagate the popular belief that

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a woman's life should be centred on her family, her home, and church affairs.It should now be evident that any attempt to improve the quality of women's

education must first and foremost focus on changing the society's attitudes towardswomen. These social attitudes which perceive women as inferior beings have, asalready noted, been institutionalized by the society within its educational system.Admittedly, mathematics and science are considered to be `male subjects' in manydeveloped countries as well. But in Papua New Guinea, not only do malesoutperform females in these subject areas, they are expected to do so. Numerousremedial measures have been taken to combat the damaging attitudes that lead tosuch expectations. Although not common in Papua New Guinea, one measurethat some countries are experimenting with is sex-segregated schooling. In PapuaNew Guinea, while most schools are coeducational, many provincial schools havesingle-sex classes in certain subject areas. For instance, it is normal practice for girlsto study home science while boys do woodwork. While it appears that girlsconsistently perform better in these gender-specific classes, it is difficult to say howmuch of this can be attributed to segregation in itself since teachers expect girlsto do well in these so-called 'female subjects'.

Mathematics Education

The majority of the population of Papua New Guinea, as noted above, lives inrural areas far removed from urban requirements of accounting, measuring andcalculating. For the most part, then, the role of mathematics in people's daily livesis limited to elementary arithmetic. It is small wonder that children from ruralbackgrounds find it hard to accept the value of abstract mathematical ideas. Addedto this, while it is true that women are officially encouraged to play an active rolein the development of Papua New Guinea, the majority of them do not even havethe opportunity to obtain a secondary education. Women pursuing higher-levelmathematics in present-day Papua New Guinea are pioneers. Naturally, this meansthey face several obstacles, among them the fact that they have few female rolemodels, their teachers being predominantly male. It is imperative, therefore, thatany strategy designed to produce gender equity in mathematics education, shouldfocus not only on encouraging more female students to take mathematics, but alsoon recruiting more female educators.

An associated problem arises because mathematics is the basis of fields such asengineering, the applied sciences, architecture and accountancy, all areas in whichfew Papua New Guinean women currently participate. Evidently, unless a womantakes an extraordinary initiative, she is not particularly encouraged to fulfill theprerequisites for securing admission to university courses in these traditionallymale subjects. Achieving gender-equity in mathematics will hopefully result inwomen having opportunities to work in these areas as well.

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Working Women

Although Papua New Guinean society is still male-dominated, change is takingplace in the cities. A few women have managed to secure professional positions asmanagers, doctors, lawyers, secondary-school teachers, and university lecturers.However, most women who have found their way into the workforce performjobs that do not require an advanced knowledge of mathematics. For instance, theyare employed as typists, receptionists, domestic labourers, preschool or elementaryteachers, nurses, and hospital orderlies. Primary-school teaching requires onlyelementary arithmetic and geometry. Secretarial work necessitates a basicknowledge of word processing and familiarity with fax machines, while jobs inbanking and the travel industry require a modest level of experience withspecialized computer packages. Nevertheless, while these jobs require somefamiliarity with technology, the related tasks are generally routine and require littlein-depth understanding of the processes involved.

Learning Mathematics

Although mathematics is often called a universal language, there are a number ofvery real problems associated with students being forced to learn mathematics ina language other than their own. In Papua New Guinea, mathematics is most oftentaught in English which, for many students, is their second, or even third,language. As a result, in trying to understand a mathematical problem, studentsmay have to resort to multiple translations. While English-speaking students caneasily translate English words directly into their mathematical equivalent, thosewho speak English as a second language often find it necessary to translate wordsinto their native language first. The situation is further complicated by the fact thatmany English concepts are quite unfamiliar to Papua New Guinean students andimportant aspects of a concept might conceivably be lost in the translation.English-language difficulties may also interfere with students' abilities to solveproblems, for example because they find it hard to translate English statements intomathematical language. When the correct mathematical interpretations areprovided, students arc usually able to perform the mathematical steps required toarrive at a solution. But, without a firm understanding of English grammaticalrules, students might have difficulty in constructing logical statements and writinganalytical explanations.

Spatial skills, visual-processing abilities, the development of time concepts,and the use of comparatives are all areas in which student weaknesses have beenidentified (Priest, 1983; Sullivan, 1982). There are many time concepts extant inPapua New Guinean culture, and several clear and unambiguous ways ofcomparing objects and situations. Unfortunately, many students do not learn theseconcepts in their own language and context since they are sent to school at a timewhen they would normally be learning these ideas at home. hi western countries,preschool teachers, as well as elementary teachers, devote considerable time and

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effort to having students use comparatives in a variety of contexts. Children alsohave access to construction toys such as 'LEGO' which help develop their spatialskills. Furthermore, teaching is based on language skills which the students alreadypossess. In contrast, Papua New Guinean students must learn the language ofinstruction at the same time they are expected to learn the mathematical concepts.In view of this, it is hardly surprising that they would develop skill in time tasksand the use of comparatives more slowly than their western counterparts.Undoubtedly, this has a substantial impact on Papua New Guinean students'development of many higher-order mathematical concepts and procedures, suchas in arithmetic, which depend on a solid understanding of these earlier concepts.A potential danger is that instructors may try to compensate for this by resortingto rote methods of teaching simply because the students have not developed asatisfactory comprehension of the prerequisite ideas. All of the foregoing mighthelp to explain w4, as has often been observed, Papua New Guinean students sitquietly in the classroom and are reluctant to participate in class discussions.

Conclusion

In this chapter I have identified a number of social and cultural issues in PapuaNew Guinea that contribute to disadvantaging girls in mathematics education.Other difficulties such as the language issues raised in the previous section, whilecommon to all students regardless of gender, serve to compound the problem forfemale students. The ques:ion at hand is how can gender differences inmathematics education be addressed?

Some social reformers have attempted to use western strategies to deal withthese problems without taking into account the deep cultural differences that existand the centuries of deeply instilled traditional values that stand in the way ofreform. For example, attempts to evoke change by using gimmicks such asmathematics medals, or competitions for girls, have resulted in complete failure.So have suggestions that the government create a separate examining board and a'female curriculum' for girls. Instead, reforms that are grounded in an under-standing that 'Rome was not built in a day' have been found to be more successful,especially in small, rural communities which resist sudden change. One suchprogramme has worked within the cultural parameters of society and gentlypresented mathematics as an interesting and viable alternative to housewifery. Thesuccess of this particular approach is yet to be evaluated.

Many politicians and educators are optimistic that women are beginning toshare equal educational opportunities. Rising tertiary enrolments and an increas-ing number of women in the workforce have been citcd as examples. As yet,however, there are no indications that the situation is improving in the rural areas,but as more schools are built one expects progress to be the natural consequence.At the same time, there is a frightening phenomenon which overshadows theruralurban debate. Law-enfbrcement offihals have noted a rising level of violenceagainst women. Even worse than the violence itself, is the fact that traditionalists

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have attributed it to the social reforms which have taken place since independence(Toft, 1985). Educational reforms have made it possible for women to have greatercontrol of their lives as well as the right to demand an education. Traditionalistsclaim it has also resulted in the loss of male authority and that the violence ismerely a way for men to protest against the disruption of their lives. The key toimproving the quality of women's lives might well be extensive legal and socialreform aimed at eliminating harmful traditional attitudes, rather than focusing onthe symptoms, namely, women's low educational qualifications.

References

BACCHUS M.K., ERI, V. and MCNAMARA, V. (1985) Report on Upper Secondary Educationin Papua New Guinea. Waigani, Department of Education.

DM'IDSON, M. and COOPF1s, C. (Eds) (1985) I'Vorking Mmen: An International Surrey, NewYork. J. Wiley and Sons.

Gu iium. G. and SMITH, P (Eds) (1980) The Education of the Papua New Guinean Child:Coqference Prom. dings, University of Papua New Guinea Publications.

KANAWI, J. (1986) Marriage in Papua New Guinea, Law Reform Commission of Papua NewGuinea, Monograph No. 4., Port Moresby, Law Reform Commission of Papua NewGuinea.

LANCY, D.E. (1983) Cross Cultural Studies in Cognition and Mathematics, New York.Academic Press.

LI.AN, D.E. (1987) Counting Systems qf Papua New Guinea, Vol. 1-17. Papua New GuineaUniversity of Technology Publications.

01 IVER, J. (1985) 'Women students at the University of Papua New Guinea', CouncilReport, University of Papua New Guinea Publications.

Pitirsi, M. (1983) 'A preparation for study in engineering and applied sciences', Researchin Matlwmatics Education in PNG.

Sui I IVAN, P. (1982) 'The development of time concepts and the use of comparatives amongPapua New Guineans'. Research in Afathematics Education in PNG.

Toi-1, S. (1985) Domestic Violence in Papua Neu' Guinea. Law Reform Commission of PapuaNew Guinea. Monograph No. 3., Port Moresby, Law Reform Commission of PapuaNew Guinea.

Wt nusiAlk E. and Cikossi I y, A. (Eds) (1988) Millen and Mathematics Education in PapuaNew Guinea, Waigani. University of Papua New Guinea Press.

-10

Chapter 17

The French Experience: The Effects ofDe-Segregation

Franfoise Delon

Introduction

France has a higher proportion of female university teachers and researchers inmathematics (24 per cent) than many other countries (Barsky et al., 1987). Thisis often explained in part by the existence of the ENSs (Ecoles Norma lesSuperieures), from which many of these teachers derive, and which until recentlywere single-sex institutions. The move to coeducation has had disastrousconsequences for the number of women studying mathematics. This chapterexplores some of the reasons for the ENSs' historical success in producing femalemathematicians and raises some concern about recent developments.

The tcoles Normales Superieure' System

The French system of higher education consists of state-funded universities withnational standards, and 'Grandes Ecoles'. The latter may be privately or publiclyfunded, possess varying entrance requirem: ].ts, and offer a variety ofprogrammes. Incomparison with universities, the 'Grandes Ecoles' prepare students for careers (forexample, political, educational, administrative, commercial) and as such, have twocomplementary roles: the reproduction of the power elite and social advancement.They recruit by means of competitive examinations for which candidates study fortwo or three years after receiving their high-school diploma. Amongst the mostprestigious of these 'Grandes Ecoles', are the ENSs which have very high entrancestandards and admission success rates varying between 5 and 8 per cent.

Students spend three or four years at the ENSs, often taking courses as wellas examinations at a local university. While the original purpose of the ENSs wassolely to produce secondary-school teachers, the range of career options nowavailable is broader, including university teaching and researchind all forms ofpublic-sector occupations especially in the senior administrative levels. In a varietyof disciplines, including mathematics, many university teachers as well as

141

Franfoise Delon

researchers conie from the ENSs. Consequently, ENS students enjoy a very highreputation in the universities and research institutions. Other advantages offered bythe ENSs to their students are excellent conditions for learning and a network ofinformation and professional relations to assist them in achieving their professionalgoals.

The Move to Coeducation

When the French State established a public, secular system of education inopposition to the existing religious one it also created the ENSs with the role oftraining teachers for the new secondary schools. The oldest ENS was created atthe rue d'Ulm in Paris and began functioning under Napoleon, at the start of thenineteenth century. Sevres and Fontenay, both for women only, and Saint-Cloud(for men) were formed at the end of the nineteenth century. The ENSET (EcoleNormale Superieure de l'Enseignement Technique), created in 1912, wascoeducational from its beginning. The ENSET, Saint-Cloud, and Fontenayschools attracted students from predominantly modest social backgrounds, whileUlm and Sevres recruited students from high schools attended only by children ofthe bourgeoisie.

With the passing of the years, the policies of the different ENSs have beenunified, although there are still noticeable differences in the social backgrounds,professional futures, and ideologies of their students. While there is no clearhierarchy among the other ENSs, Ulm, traditionally a men's school (with somenuances, see Hulin, 1994), is definitely the most prestigious. Several famouspoliticians, writers, and scientists are alumni of Ulm, and the institution pridesitself on being the route to a brilliant career. It is considered the 'temple' ofintellect and academia. The situation at Sevres has evolved over time (Mayeur,1994; Hulin, 1994), and has always been ambiguous. Despite sharing its entranceexaminations and official statutes with Ulm since 1936, the faculty and alumni atSevres are not as powerfully established in the university community. Indeed in1981, at the celebration of the centenary of Sevres, while acknowledging that theENS at Sevres had accomplished part of its role in that women now have accessto knowledge, several officials pointed out that women had yet to accede to power.Whether they were referring to the male-dominated world of politics, or to powerin the intellectual world is not clear. However, what is certain is that they appearedto believe that women had no further need for the protective structure of thewomen's ENSs and, in fact, were perhaps even disadvantaged because of thepoorer reputation of their schools. Soon after this event followed first the mergerof Fontenay and Saint-Cloud, and then Sevres and Ulm, without any serious prioranalysis as to thc possible consequences. The mergers did not prove to befivourable to women in any discipline (Ferrand, 1994). However, in no subjectwere the consequences as disastrous as in the area of mathematics, and at this pointit would be appropriate to turn to a consideration of this phenomenon.

142

Tile French Experience: The Effects qf De-Segregation

The Situation after Coeducation

The mathematics requirements of the entrance examinations for the individualENSs were all identical. Furthermore, within each of the pairs, Sevres/Ulm andFontenay/Saint-Cloud, the written examinations were conmion, although assess-ment was done independently. In the past, each institution had a certain numberof designated places for mathematics, about forty for the men's schools and abouttwenty for the women's schools (fewer females applied). In the eight years sincethe merger of Sevres and Ulm, only twenty-one females have entered mathe-matics. According to Ferrand, Imbert, and Marry (1993) this amounts to a declinein the numbers of female mathematicians in these years of over 80 per cent.

Several factors contribute to ,this alarming statistic. Within the preparatoryclasses for the scientific 'Grandes Ecoles', between 15 and 20 per cent of studentsin the most mathematically challenging sections are women (Barsky et al., 1987).However, the corresponding figure stands at only 10 per cent in the mostprestigious high schools which furnish the majority of ENS students. Here onefinds the most dramatic form of the lingering estrangement that girls demonstratewhere mathematics is concerned throughout their schoolthg. They ofien attributetheir reluctance towards pursuing the subject to its 'abstraction or its remotenessfrom human concerns (Mathematiques A Venir--Opération '50 lycées', 1988). Itis nnportant to realize, however, that in France mathematics is used as a gatekeeperto higher education and that an interest in mathematics conveys as much a concernfor a professional future as a taste for the subject itself (Baudelot and Establet, 1992;Duru-Bellat, 199(f).

Mathematics is, in fact, the key to the most prestigious professions. Thisphenomenon. which holds true throughout schooling, increases in impact in theclasses which prepare students for the ENSs. The work required in this setting is moreintense than at any other time in a student's education. It is highly structured andfocused on its ultimate goal, that ofadmission to one ofthe schools which function asa springboard to employment at the top of the public sector and decision-makingprokssions, a world that is still almost exclusively masculine. Many families,particularly those belonging to the upper social classes, expect their male offspringtohave professional ambitions at an early age and plan their studies to this end. At thesame time, they are often less concerned with the education of their female children.Prepared and supported by familial and societal expectations, boys tend to accept theconstraints ofpreparatory classes more readily than girls. In addition, they also appearmore comfortable with the competitive atmosphere found there.

This is the social context of the preparatory classes and one which functionsto assist boys more than girls in Coll( entrating and mobilizing their energies. Thereis, howeverinother, aspect of the preparatory classes which should not be ignored.The oldest and most prestigious of them, and hence the ones which provide thegreatest chance of success, have well-established rituals including their own specialvocabulary% private jokes. initiation rites, and a tendency to tbrm hierarchies, allof which might seem more in keeping \vith military school. There is a strongpressure on students to conform to the 'spirit' of the preparatory classes, and any

14.3

Françoise Delon

resistance to this might be a source of difficulty for the student. Given theirminority status in such an environment, women are often victimized and made thetarget of remarks of astonishing rudeness. (For an account of the 'initiation rites'in the preparatory classes and their discriminatory content, see Dupe, 1992.)

Thus, the preparatory classes do not constitute a particularly enjoyableexperience, especially for women; solid motivation and a reasonable expectationof success are essential for those who commit themselves to attending them. Beforethe mergers, the women-only ENSs represented a reassuring alternative fromseveral viewpoints: their existence allowed women to escape the severest forms ofcompetition, and offered honourable and reasonable professional prospects. Afterall, were they not destined at the end of their ENS studies to obtain that mostfemale of occupations secondary school teaching? However, the professionalexpectations of the successful were changed by the new environment. In the ENSsthere are greater expectations that students will pursue professions other than high-school teaching. The women-only ENSs made possible a smooth transition fromtraditional expectations to more career-oriented ones. This route is now moredifficult for women. The finnale students of the mathematics preparatory classesknow it well, for they apply in much smaller numbers to the coeducational ENSthan they did to Sevres. One may fear that the dePeto closing of the ENSs to femalemathematicians will push many female students out of the preparatory classes, withonly those with early established professional goals remaining.

Women's 'Ecoles Normales Superieures'

Let's return again to an examination of the women's ENSs. Before coeducation,the ENSs had five to ten times the number of female mathematics students thanthey do today. Traditionally. women's ENSs have produced a large number ofhighly qualified teachers who enjoy a great intellectual prestige, and as such haveprovided generations of high-school girls with female academic role-models.These institutions have spawned many of the regrettably few scientificallyrenowned female figuresis well as facilitated the establishment of women in theuniversity sphere (Coste-Roy, 1987; Barsky et al., 1987). The women-onlyenvironment, which was both reassuring and stimulating, gave women anintellectual legitimacy which enabled them to enter mixed competitive environ-ments such as professional recruitment examinations. Even if they met with lessSuccess than their male counterparts, the proportion of women who did achieveSuccess Was greater than would be expected given their actual numbers in thepreparatory classes (Imbert. 1994). This world of women had the flavour of anutopia. Women's ENSs provided an image of intellectual women that was quitedifferent from the stereotypical one, that being mostly single women enjoyingbooks, test-tubes, courses, journeys, and meetings. Whether this new image hadany validity is unimportant. But it was the myth, and a myth whose value by inthe provision of an image counter to that of the housebound wife withoutemotional autonomy, or the superwoman, seductive and successful in both career

144

The Frem It Everieme. The Efli'a of De-Segn.gatson

and family. The existence of this so-called 'other' exposed he contradictions andunrealistic dimensions of these misconceptions. It made it possible to imagine rolesother than those provided in advance. It provided an environment where nobodywould have thought of describing Emmy Noether as 'fat. rough, and loud'. l Mythsaside, the women's ENSs were, for some of their students, crucial places for theacquisition of personal maturity and autonomy. It was at the ENS where visionsof certain female figures became ingrained in the minds of students, helping themto establish themselves in the world as women.

Conclusion

The question now becomes what will happen next? If present-day figuresconcerning women mathematicians at the ENSs do not change, one may fear asignificant decrease in the number of women in research and university positionsin the future. The women's ENSs played a crucial role in promoting women inmathematics. They produced most of the pioneers and, up until recently a sizeableportion of female researchers and university teachers, particularly those in seniorpositions. The demise of the single-sex ENSs came in the name of modernity oreven in the name of equity. The illusion consisted of thinking that coeducationalinstitutions could be egalitarian in a world which is not.2 The question, therefore,is how to remedy this? The idea of setting quotas is not popular in France.Furthermore, in the present context it is seen as contradicting the aura ofobjectivity attributed to mathematics, and to the principle of selection bycompetitive examination. Perhaps it will be possible to profit from EuropeanCommunity structures which permit affirmative action to offset historicaldiscriniination. An analysis of current recruitment procedures may shed light onnew possibilities, besides being healthy for the profession as a whole.

In the present system of preparatory classes and competitive examing thesuccess of some is only possible because of the failure of others. The fact that manymathematicians are the product of such a puocess probably explains the lack ofsolidarity sometimes observed in the French mathematical community Inaddition, the rigidity of such an environment favours the emergence ofmathematicians with remarkably homogeneous intellectual and social behaviours.There may be much to gain by diversifying modes of training, criteria, age ofrecruitment, and ways of working. Thb approach might be preferable to theartificial promotion of particularly rigid careers or scientific profiles. Moreover.women may tind thenbelves less torn between contradicting stereotypes if a moreflexible system and ideology were adopted.

Notes

On a poster depicting nuthematicians throughout history. Emmy Noether the onlywoman appearing is so des( ribed. This poster hangs in many mathematicsdepartments and universities in North Ainerica.

1 4 5

Frattwise Delott

There are no legal obstacles in France to employment equity. Theoretic,* allprofessions are open to women. Three-fourths of women of working age do work,but, on thc average, they are disidvantaged with respect to men in terms of salary.advancement. and employment.

References

BR,,RY, D.. CI IAI . M.. CI IRIS , G., Goinin I ON, C.. KI I IN, E..

J.-Y.. MOIGI IN, C., OvAl RI, J.-L.. RAOLVI. A., RoussitiNoi M. andSIMON, J. (1987) 'Demographic des mathematiciens. Des mathematiques findustrie.La fuite des cerv,iix', Bulletin de la Soth'u: .11atUmatique de France. Supplement au tome115, pp. 343-91.

13.m..1iut oi, C. and Es IABI ii. R. (1992) AM.:: les Fab, Paris, Seuil.Cos ii -RtlY, M.-F. (1987) 'La place des femmes dans les mathematiques: Probkmes actuels.

perspectives d'avenir'. Bulletin de la Socii.k: Matlu'tnatique de France. Supplement au tome115. pp. 320-42 (in particular the contributions of). Ferrand, N. Desolneux and D.Perrin)., M.-0. (Ed) (1992) Biz-utago. Conde-sur-Noireau, Arka-Corlet-Panoramique n" 6.

Dt.ku-Bi I I. M. (1990) L'Ecoh. des Filks, Paris. l'Harmattan.Ft RRANI ) J. (1994) 'Une generation sacrifke', St'vriennes d'Ilier et d'.-lujourd'hui, 148 (dec.

93-mars 94), pp. 15-17., M., IMIII is I. F. and MARRY. C. (1993) 'Normaliens. normaliennes scientifiques:

l'excellence a-t-elle un sexe?'. communication. colloque Pour nit nouveau bilan de laAiciologie. Paris, Institut National de la Recherche Pedagogique, 25-27 mai 1993.

H1.1 IN, N. (1994) 'La section de. s sciencec de l'Ecole Normale Superieure: Quelques jalonsde son histoire', in SIRINI I I I. J.-F. (Ed) Licole Normal(' SupiTieure. Le Livre Sn Bicentenaire,

Paris. Presses Universitaires de France, pp. 321-49.IMBI Isl. F. (1994) 'Agregations scientitiques et ENS au temps de la mixite'. Unpublished

manuscript..Mathematiques A Venir Operation '50 lycees' (1988) LA'5 Maths et I;nts, Strasbourg,

Institut de Recherche sur l'Enseignement des Mathematiques.MAYI t 'Is. E (1994) 'Severs', in Silsisa iii, J.-F. (Ed) Ecole Siqu'rieure. Livre du

&immure, Paris. Presses Universitaires de France. pp. 73-111.

146

Chapter 18

The Contribution of Girls-OnlySchools to Mathematics and ScienceEducation in Malawi

Pat Hiddleston

Introduction

Malawi is a small. land-locked, poverty-stricken country in Central Africa. Evenby African standards Malawi is densely populated, with one million Mozambicanrefugees in addition to its own nine million peopk. The local economy isprimarily based on agriculture. and as a result, rural settlement is the predominantpattern. In comparison with the western world, the country lacks technologicaladvantages such as television and calculators: dt school, for example. tables oflogarithms are still quite commonly in use, The rate of participation in theeducational system drops at each successive level. For instanceiccording toMinistry of Education and Culture (1990) statistics only 41.7 per cent of prnnary-school-aged children attend schooli figure which drops to only 3.4 per cent atthe secondary levelind thereafter less than 1 per cent receive any form of tertiaryeducation (Ministry of Education and Culture. 1990). Entry to secondary schoolis determined by means of a highly selective examination. the PSLE (PrimarySchool Leaving Examination). A Ministry of Education affirmative action policyrequires that one-third of all secondary-school places be reserved for girls, but inorder to achieve this quota, many girls are admitted to school with lower PSLEscores than is characteristic of the boys.

Malawi's only university has five colleges, the largest of which. ChancellorCollege, incorporates faculties of science, humanities. sociology, law, and educa-tion. Admission to Chancellor College is based on grades in the final secondary-school exaiMnation known as the MCE (Malawi Certificate of Education). Girlsaccount for only 23 per cent of the student population of Chancellor College.despite the fact that they are granted entry to university with lower scores thanboys and comprise 34 per cent of the university student population as a whole(World Bank. 1988: Chancellor College, 1991).

Until recently, there has been very little research on women's issues in thedeveloping world. In the few caws where such research has been undertaken, it

147

Pat Hiddleston

has been conducted in isolation and has been difficult to publish. Yet over 75 percent of the world's women live in developing countries. Since the needs of thesewomen are often quite different from women in the developed world, research ondeveloped nations cannot be assumed to apply universally. This chapter contributesto the growing body of literature on gender issues in the developing world byelaborating on several salient issues concerning students at Chancellor College.First, the majority of female students at Chancellor College attended girls-onlyschools and achieved higher MCE scores than their counterparts from coeduca-tional facilities. Second, while female students entering Chancellor College do sowith relatively low grades compared with males, especially in mathematics, they

improve to a much greater extent and graduate with results that are, on average,higher than those of the male students.

Girls-Only Schools and University Selection

Developing countries have not been the target of much investigation concerningthe relative merits of single-sex and coeducational schooling. However, independ-ent research conducted both in Nigeria and in Thailand points to single-sexschooling as providing the most beneficial learning experience for girls (Lee andLockhead, 1990; Jimenez and Lockhead. 1989). There are three categories ofschools in Malawi: government, grant-aided, and private. The government schoolsmay be either single-sex or coeducational, although the newer ones are mostly thelatter. Many single-sex schools are grant-aided, and while they enjoy a certaindegree of freedom from Ministry of Education regulations. they are still compelledto adopt the same selection procedures and charge the same fees as governmentschools. Therefore, in most respects these two types of educational facility aresimilar. There are so few private schools that they may be disregarded with littleimpact on the analysis.

Table 18.1 illustrates the performance of girls in the MCE examination for theyears 1987 to 1991. It shows that the percentage of MCE passes in each year wasconsistently higher in the girls-only schools. At the same time, the coeducationalschools were seeing a much greater increase in the number of girls taking theexamination, undoubtedly due to the newer government schools which werepredominantly coeducational. The Ministry of Education insists that schoolselection is independent of the academic standing of the students and that there isno bias towards placing the best girls in single-sex schools.

If there is any validity in the Ministry of Education's claim that they do notplace the brighter students in single-sex schools, one would expect the numbersof girls admitted to Chancellor College from both types of institution to beroughly in proportion to the numbers who pass the MCE. Table 18.2, however.shows that from 1987--1991, the number of girls entering Chancellor Collegefrom girls-only schools was substantially higher. This pattern persists when onerestricts the data to those entering the two science-based programmes, or indeedany undergraduate programmemd it has not changed as the intake from

148

The Contribution of Girls-Only Schools

Table 18.1: MCE performance of girls from coeducational and girl-only schools for 1987-1991

Coeducational

Sat Passed Percentage

Girls-only

Sat Passed Percentage

1987 510 253 49.6 901 643 71.41988 583 305 52.3 1067 722 67.71989 1026 471 45.9 1073 599 55.81990 1106 452 40.9 1294 760 58.71991 1068 414 38.8 1260 745 59.1

Source: Chancellor College c19911 Statistics

Table 18.2. Intake of girls to Chancellor College. as a percentage of total intake, for all degreecourses by type of school

Total intake

Girls as a Percentagepercentage of girls fromtotal intake coed schools

Percentagegirls fromgirls-onlyschools

Percentgegirls from

other schools

1986-87 227 17 1 13 31987-88 293 14 3 10 1

1988-89 278 24 3 18 31989-90 329 18 3 13 21990-91 379 23 4 17 21991-92 409 19 4 12 3

Source Chancellor College (19911 Statistics

coeducational schools has increased. All in all, the situation in Malawi educationappears to be consistent with that in both Nigeria and Thailand, that is. studentsfront female-only schools meet with greater academic success than their coeduca-tional sisters.

Gender Comparisons of Performance

At all stages of their school careers, girls in Malawi are outperformed by their malecounterparts in mathematics. In fact, PSLE scores for the years 1989 to 1991indicate that primary-school boys achieved better results than girls in all subjectareas (Bradbury, 1991). Kadzamira (1987) has documented a similar trend for high-school students on the MCE in the period 1982 to 1986. As previously discussed,affirmative action policies result in the admission to Chancellor College ofwomenwith qualifications that are comparatively lower than those amen. Moreover, thesituation is particularly acute in the area of mathematics. However, when oneexAmines performance in the first-year mathematics course, one notes a substantialimprovement in female performai ice over the year relative to males.

Table 18.3 documents this trend for the years 1987 to 1991 . It shows women's

149

Pat Hiddleston

Table 18.3: First-year mathematics grades at Chancellor College from 1987 to 1991

FinalTotal MCE MCE Test 1 Test 2 Test 3 Exam

Total female (all) Maths Nov Feb Mar June

87/88 223 25 50.2 45.5 49 5 49.4 49.2 45.2

88/89 175 29 47.9 '46.1. 49.0 50.1 51.6 51 8

89/90 206 32 46.3 45.8 43.2 47 3 49.1 50.2

90/91 258 55 45.5 43.4 45.2 46.2 47.0 47.7

Average 47.0 44 9 46.3 47.8 48.8_

48 7

Source: Chancellor College (1991) Statistics

Table 18.4 Second-year mathematics grades at Chancellor College from 1988 to 1992

MCE Year One Year Two. .

Total F All Math T1 T2 T3 Ex T1 T2 Ex

88/89 136 11 52 4 47 5 52.3 54.1 51.0 50 5 48.8 52 3 49.6

89/90 100 16 47 4 46 9 48.3 48 5 51.1 51 7 - 54 4

90/91 115 14 49.8 47.7 43 5 50 4 51.6 52 6 51.3 62.6 52.0

91/92 141 26 44.1 40 6 45 6.,' 46 1 45.5 47.6 45.0 47.9 49 8

Average 47.4 44 7 46 9 48 9 49 0 50 1 47 5 50.1 1 3

Source Chancellor College (1991) Statistics

scores, standardized to a mean of 50 for the entire cohort, for three term tests aswell as for the final exannnation. Taking an average of all the years shown, thegrade improvement throughout the course is quite apparent. The fact that all

students were taught and evaluated by the same (male) lecturer, ensured the

necessary level of controlTable 18.4 documents continued improvement into the second year, with the

data for first-year students adjusted to include only those who did, indeed, advanceto take the second-year mathematics course. It is interesting to note that the femaleaverage for all years shown of 51.3 on the second-year final exam exceeds thecorresponding figure of 50 for the whole group. In essence, this means that oncompletion of their second mathematics course, female students, on average.achieve at least as well as, if not better than, their male counterparts. (Again, theconditions were controlled in the same manlier as before.)

This trend of improved performance is not confined to the first two years ofmathematics, but extends to the sciences as well as into more advancedmathematics courses. Table 18.5 shows average female-standard (Z) scores for therears 1981 to 1990. Seven subject areas are included, biology, chemistry, computerscience. earth science, mathematics. physicsmd statistics. Computer science is notoffered in year one, while statistics is offered only in years three and four. In

150

The Contribution of Girls-Only Schools

Table 18.5. Z Scores for end-of-year grades in the Science Faculty, Chancellor College, by yearof study

Computer EarthBiology Chemistry Science Science Maths Physics Stats

1 0.38 0.27 0.24 -0.36 -0.542 0.30 0.00 0.26 -0.40 0.05 -0.453 0.27 0.06 0.96 0.08 0.17 0 32 0.39 0.68 1.19 0.01 0 93 -0.10 0.15 -0 134 0.22 0.34 0.27 0.43 0.05 0.36 0.52 0.11 1.35 0 36 0 94 0 45 1.81 0.82 0.02

Source: Chancellor College (1991) Statistics

addition, as is illustrated in the table, certain disciplines have more than one courseat sonic levels.

The data presented in the table for year one shows that all subject areas, apartfrom earth science, have negative Z scores. This means that female studentsgenerally perform below average in the indicated courses. However, as oneprogresses through each of the years, the overall trend is for the scores to graduallyincrease until they are all positive. Thus, by year four, female students are achievingabove average in all documented subjects. These findings agree with those ofMaritim (1985), whose study of Kenyan high-schools concluded that girls withlower high-school entry grades appear to outperform boys in their finalmathematics examinations. These results contrast, however, with the majority ofresearch findings in the developed world. In developed countries, male academicperformance tends to surpass that of females in early adolescence, a trend whichcontinues well into later years (Burton, 1990).

Several factors may combine to explain women.s apparent success atChancellor College. First, while Malawian school teachers expect girls to havelower grades than boys in mathematics and science, the staff at Chancellor Collegedo not share this opinion. Indeed, they have no preconceived expectations.Second, women who enter the college experience a sense of personal achievementoil having done so, and claim no longer to doubt their abilities. Thus, there is ashift from a self-fulfilling expectation of failure, to the anticipation of success.Third, women at Chancellor College have a reputation for working harder thanmen, possibly as has been suggested. because they feel obliged to justify beingadmitted in the first place. Finally, Chancellor College is a residential institution.As a result, women are able to concentrate on their studies without having tocontend with domestic chores. In contrast, school girls are often forced to beabsent from school for domestic reasons, expectation that does not hold forboys.

Conclusion

Females in Malawi who attend girls-only schools achieve better results on the finalschool examination than those who attend coeducational institutions. This is true

15 1t.)

Pat Hiddleston

even though they are sekcted for these schools by a process which, the Ministryof Education claims, has no bias towards placing the more academically successful

girls in single-sex learning environments. Far more women are admitted toChancellor College from girls-only schools, than would be expected from theproportion qualifying. While they enter the university with lower qualifications

than men, women's results in the mathematics and sciences improve substantially.

By the time they graduate, females outperform their male counterparts in these

subject areas.

References

i3RAI nit ity. R. (1990) Difi-ert.nces BeniVo the Results of Boys ,nul Girls in Sub.jeas (9`. the Primary

School I..eal.ing Examination. Malawi Examination Board, RTC 14/90.BUR. I oN, L. (1990) (;ender and Mathematics: An International Perspective. London, Cassell,

IANCI I I Olt. COI I I co (1991) Statistics.JIN11 \ /, E. and L., )( II AI M.E. (1989) lhe Relative Effectiveness Sinsge Sex Schools in

Thailand. Educational and Policy Analysis.K.s0./Amiu./. M. (1987) Sex DiffiTences in the Perfinmance of Candidates in the AlAnvi

Certificate of Education in Alathetnatics and Science Sul!jects. Malawi Examination Board,

RTC 21/87.Li I . VE. and L(1( I:111 AI M.E. (1990) 'The effects of singk sex schooling on achievement

and attitudes in Nigeria'. Comparatiii Education Review 34.2, pp. 209-31.MARI I INS. E.E. (1985) 'The dependence of "CY and "A- level results on the sex of the

examinees'. Kenya Journal of Education. 2, pp. 21-46.MINI', I RV ( )1 El cii A I It ANI Cl I Lit' (1990) Education Statisno. Lilongwe. Malawi

Government.ntii) BA Ni (1988) Education in Sub-Sahaia Altka. .for Ail.lionnent. Revitalisation,

and Expansion, Washington. DC, W(crld Bank.

15 )

MN,

Part 3

Feminist Pedagogy in MathematicsEducation

The fourth phase of gender reform of mathematics education, described in theintroduction asks the question: What would mathematics and mathematicseducation look like if women were central to its development? We have dividedthe chapters in Part 3 into two categories according to whether they emphasizechanging mathematics pedagogy or changing the discipline. Although these departfrom different standpoints, they come to similar conclusions and lead us tospeculate as to the nature of an inclusive mathematics education.

The influence of feminism on approaches for achieving equity in mathematicseducation has perhaps been most evident in the area of pedagogy. Sonw of thechapters in this section provide a theoretical description of a pedagogy based onan understanding of women's ways of knowing, others give practical suggestionsfor implementing a feminist mathematics pedagogy. Whatever their focus, all sharethe desire to see a fundamental change in the distribution of power in theclassroom and as a consequence in the organization of the discipline ofmathematics as a whole.

In the first chapter, Roberta Mura defines three so Inds within contemporaryfeminism and demonstrates how all approaches to gender reform of mathematicseducation are, to different extents, influenced by soni,. form of feminism. Thechapter by Joanne Rossi Becker describes stages in women's ways of knowing, theconnected teaching that grows out of an understanding of this modelmdexamples of how to apply this theory to mathematics. The practical implementa-tion of a feminist pedagogy is the focus of the chapter by Pat Rogers. Drawing onGilligan's theory of gender-related differences in moral development, shedescribes how she abandoned the authoritative lecture format which silenceswomen and began developing a creative-intuitive pedagogy that has beenparticularly successful with her female undergraduates. Sue Willis, based on afeminist critique of pi evious proposals to change the gender imbalance in schoolmathematics, develops proposals for a different madwmatics curriculum, onewhich is gender-inclusive and socially critical. This impliesunong other things,non -sexist curriculum content, the use of a variety of intuitive and exploratorypploac hes to mathematics, connecting mathematics to reality, and the explicit«mfrontmg in the classroom of gender issues in the wider community. The focusof Joanna Higgins' chapter is the use made by teachers of games as Independent,i t. tis mes in the teaching and learning of mathenutics at the elementary school

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Feminist Pedagogy in Alathematics Education

level. She argues that the classroom experiences of girls differ from those of boysin ways that disadvantage the girls and so the structures of learning experiences ifleft unexamined tend to benefit boys more than girls.

The chapters of Leone Burton, Betty Johnston and Marjolijn Witte, includedin the second group, all deal with the need to change the discipline of mathematics.They ask whether there exists a female mathematics and whether it would bedifferent from mathematics as developed thus far. Leone Burton, referring to theextensive literature on gender and the nature of science, asks similar philosophical,pedagogical and epistemological questions about the discipline of mathematics.Betty Johnston reflects instead on the interaction between mathematics andsiety. in particular on how the quantification of society constructs us and howwe, in turn, construct or resist it. Through memory work, she explores usingpersonal experience as the basis of knowledge and examines the interface betweenthe everyday world and wider social structures. One particular thread that emergesis the experience of mathematics as regulation by others, a thread that is also pickedup by Marjolijn Witte. Distinguishing several difkrent strategies for intervening toprevent girls from dropping out of mathematics education. Witte explains how thedominant language in mathematics constrains the actions of learners anddisadvantages girls. Within this language conception of mathematics, she discrim-inates two basic conceptions of mathematics, one of which she claims holdspromise for girls.

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Chapter 19

Feminism and Strategies for RedressingGender Imbalance in Mathematics

Roberta Alura

A variety of theoretical perspectives has been adv,w::::d for addressing the issue ofgender imbalance in the teaching and learning of mathematics. For instance, inKaiser-Messmer and Rogers (1994), four examples are given: 'The "interventionperspective" which locates the problem in the student; the "segregation per-spective" in which the interaction between girls and boys is often the primaryfocus; the "discipline perspective" in which the nature of mathematics itself isproblematized; and the "feminist perspective" in which a critique of the genderednature of teaching and learning mathematics leads to an examination of the use Ofpower and authority in the mathematics community' (Ibid., p. 304). For me, thisraises some questions. Why did the authors label only one of these tburperspectis es feminist? Don't all approaches which seek to address the imbalancecorrespond to some kind of feminist anal).sis and practice? Indeed. how doesfeminism relate to work on women and mathematics?

Although the word 'feminism' usually occurs in the singular, there are in facta variety of feminist theories. In this chapter, I consider the possible links betweendifferent kinds of feminisms and the different approaches to the issue of genderimbalance in mathematics education described above and illustrated by fourselected positions, namely, those of Sheila Tobias (1978), Charlene and JamesMorrow Pat Rogers and Sue Willis (see Chapters 2, 21 and 22, respectively, thisvolume). I begin with a brief discussion of feminist them v.

Trends within Feminism

It is likely that means something difkrent to every feminist. andsomething else again to those who oppose it. All feminists probably would agreethat women suffer certain disadvantages ill comparison with men. Beyond that,they might analyse the situation differently and favour different strategies forchanging it. In this section. 1 draw on a typology proposed by Descarries-Bdangerand Roy (1991) to identify three main trends within contemporary feminism:

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'feminism of 'radical feminism', and 'feminism of difference' (nohierarchy is implied by the order used here))

Feminism of equality is action-oriented; it demands legal and actual equalitybetween women and men, and it identifies the sexual division of labour as themain source of women's oppression. It denounces, first and foremost, thediscriminatory conditions women experience in the spheres of education andwork, and advocates programmes that would ensure equal access for women to allsocial, political and economic resources. Together with political action, thesocialization and education 'girls are seen as the most effective tools for change.Of the three feminist currents discussed here, this is the one that conies closest tothe typical definition of feminism found in English and French dictionaries.Feminifts of equality might extend their criticism of sexual discrimination to allother kinds of discrimination, however they do not necessarily challenge socialinstitutions, they ask 'only' that women occupy 50 per cent of all political officesas well as all military, religious, academic, managerial, judicial, and other offices.Some consider feminism of equality reformist, while others think that trueequality between the sexes in all social, political and economic spheres would betantamount to a complete revolution.

Radical feminists refuse to define themselves in comparison with men.2 Theyidentify patriarchy as a social, political and economic system that oppresses andexploits wonwn individually and collectively, sexually and economically. Radical

minists have put more energy than feminists of equality into theoreticalproductions and their positions are also more diversified. Descarrie -Belanger andRoy (1991) distinguish three tendencies within the radical current, namely,materialist feminism, 'woman-centred' feminism, and lesbian feminism. I won'tdwell on these distinctions here. Radical feminists see patriarchy as characterizedby hierarchical relations between the sexes, maintained through social institutioLslike marriage and heterosexuality, through laws, religion and the educationalsystem, as well as through violence or the threat of violence. Radical feminismrefuses the separation between domestic and public life and questions theandrocentric perspective that pervades all aspects of patriarchal cultures. Strategi-cally, priority is given to ending all types of violence against women and to actionsaimed at the disappearance of thc institutions responsible for their oppression andexploitation. Radical feminists might extend their criticism of the oppression ofwomen to all other kinds of oppression.

Unlike feminists of equality and radical femMists, feminists of difference donot wish to eliminate gender distinc,ons. On the contrary, they insist on therecognition of difference, t!,ey see women as possessors of specific knowledge,culture and experiences and the feminine as an affirmation oflife. They assert thatwomen have their own ethics, their own way of knowing and their own language.Their project involves reclaiiMng and revaluing all things feminine. Influenced bypsychoanalytical theory, they want to discover and liberate a truly femininesexuality that has been repressed by patriarchal culture. Since women's lecificityconsists principally in their childbearing and nurturing capacities. nm. rhood isat the heart of feminism ofdifference. Feminists of difference demand a better deal

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Feminism and Strategies ji)r Redressim Gender Imbalance in Matlumatics

for wives and mothersind are fierce opponents of the new reproductivetechnologies which they see as threatening to take away from women their controlof motherhood and undermining their very definition. Feminists of differencepromote the idea of the complementarity of the sexes and ask that women beallowed to engage in activities separate and different from Alen, but that theseactivities be equally valued and rewarded.

As with all classifications, this one distorts reality: individual feminists mayhold ideas and participate in actions typical of two or all three trends. However.I believe that these categories reflect real differences within the women'smovement. Sometimes, they are just a matter of focus and priorities, butsometimes they correspond to serious disagreements, where the proponents ofopposing theories would deny each other the right to call themselves feminists.

Redressing the Gender Imbalance in Mathematics

What connections can be found between the three feminist currents describedabove and the perspectives on gender imbalance in mathematics discussed earlier?Clem. ly. redressing thc gender imbalance in the teaching and learning ofmathematics is part of a more global project of achieving educational andoccupational equity between the sexes and is in itself a typical feminism-of-equality issue. Therefore, this current underlies all approaches seeking to redressimbalances. The intervention perspective, though, is especially characteristic ofthis brand of feminism, because of its emphasis on increasing the participation ofwomen in mathematics and on programmes aimed at resocializing girls, that is,correcting, or compensating for, a deficient socialization. The early work of Tobias(1978) illustrates this perspective well, by espousing the goal of getting girls tochoose mathematics and to persist in it, by focusing on an attitude of the individualstudent, namely mathematics anxiety, as the source of her mathematics avoidance,b; assuming that this attitude is learned and therefore changeable, and by actingto help students unlearn it.

In contrast, the segregation perspective, is reminiscent of feminism ofdifference, in that it asserts that boys and girls have different ways of learning andthat they are better taught separately, using methods and/or curricula appropriateto each. Whatever the origin of gender differences in learning styles, proponentsof the segregation perspective considet it a given, either unchangeable or worthpreserving. Support for this view is found in research like that of Belenky et al.(1986).

The Morrows (see Chapter 2, this volume) reconmiend the segregationoption on other grounds. l'hey focus on the interaction between girls and boys.In a mixed group, they argue, boys always end up getting a greaier share of theavailable resources than girls. Although ihe Morrows attribute this genderdifference in behaviour to socialization, they advocate segregation as a way offreeing girls from the oppressive situation that prevails in coeducational environ-ments, for changes in socialization, nnich as they are desirable, would consume

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more time and energy than teachers and students usually can afford duringmathematics classes. This argument proceeds from a radical feminist standpointwhich analyses classroom interactions in terms of oppression and dominance andaims at girls' autonomous development without reference to boys..5 It claimsneither equality nor difference.

The discipline perspective is described above as the view that it is the discipline ofmathematics itself that leads to the gender imbalance. Both radical feminism andfeminism of difference may be relevant to this view. The former has launched anambitious project of examining all existing knowledge for evidence of androcen-trism. In spite of the apparent difficulty of the task, one may want to submitmathematics and the physical sciences to the same kind of feminist criticism that isbeing applied to other disciplines. Feminists of difference, furthermore, mayentertain the hypothesis that mathematics, as it presently exists, is the product of amale way of thinking and may speculate about the possibility of a femalemathematics. These ideas, however, are regarded with suspicion and hostility bymathematicians subscribing to feminism ofequality, some ofwhom fear that feministcritiques of science may discourage women from entering the field.' I believe thisfear to be unwarranted since several disciplines, such as biology, psychology, andliterature, at least iii Canada, have higher female participation rates than mathematics,despite being the target of much harsher feminist critiques.

Willis (see Chapter 22, this volume) espouses the discipline perspective, butin doing so she focuses on school mathematics and on the image of mathematicsrather Jan on the discipline itseif Of course, one could argue that schoolmathematics is part of mathematics, however Burton (1987) points out that thefeatures of school mathematics singled out as potentially responsible for the genderimbalance are precisely those that distinguish it from the discipline as actuallypractised.

The feminist (the fourth) perspective, discussed earlier places responsibility forthe gender imbalance on the teaching of mathematics and calls tbr improvedteaching methods that will benefit male and female students alike. All threefeminist trends, not to mention the whole mathematics education communit\ .

have an interest in this perspective. Fenfinists of equality demand non-sexistteaching, that is, fair treatment of female and male students. Feminists of differencespecify that fairness must take into account difference, and that equal treatmentdoes not necessarily constitute equity. They insist on a pedagogy consonant withwomen's psychology, a pedagogy sensitive to students' emotions and based oncollaboration rather than competition. Radical feminists favour a pedagogy thataddresses the subject of women's oppression and liberation explicitly in theclassroom. ln line with radical thinking, this pedagogy seeks to minimizehierarchical relations between teacher and students, empowering female students.and forcing male students to give up their dominance in the classroom. Asmentioned earlier, this typ. of feminist analysis of what goes on in the mathematicsclassroom naturally 'leads to an ex,nnination of the use of power and authority inthe mathematics community' (Kaiser-Messmer and Rogers, 1994, p. 304)indbeyond.

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Feminism and Strategies for Redressing Gender hnbalame in Mathemath-s

Both Rogers and Willis (Chapters 21 and 22, this volume) advocate a radicalfeminist pedagogical perspective and they both extend it to a more inclusivepedagogy of liberation, drawing on Freire's ideas (1971), exposing the role playedby mathematics in constructing and maintaining privilege, be it gender, class orrace privilege (see also Frankenstein, 1983, 1991, and Rogers, 1990).

It seems to me that the theoretical perspectives for redressing the genderimbalance in mathematics that have been outlined above could more precisely becalled strategies, and that three emerge from the approaches discussed: theintervention strategy the segregation strategy and the teaching strategy. The laststrategy corresponds to the perspective identified as 'feminist' by Kaiser-Messmerand Rogers and incorporates aspects of the discipline perspective in that itconcerns both pedagogy and curriculum. I have argued that each of thesestrategies can be linked to at least one feminist current. That all three can lay claimto feminism follows from their common intent to improve the situationof womenin mathematics.

The claim to feminism of all three strategies does not preclude criticism of anyof them for being less than effective, or even for being counter-productive. Sucha debate parallels the one about the difftrent tendencies within feminism and theirpotential for improving the situation of women in society. For instance, one couldfind fault with the intervention strategy for laying the responsibility for redressingthe balance at the feet of girls and women (that is, for assuming that it is they whohave to change) and for failing to criticize mathematics, mathematics education,or the educational system as presently configured. Like the feminism of equalityunderlying it, the intervention strategy may seem to encourage women to reachfor half the pie without stopping to consider whether the pie is worth eating.

The segregation and the teaching strategies, depending on whether they drawon feminism of difference or on radical feminism, may be criticized for eitherreinforcing gender stereotypes (women and men think and learn differently)', orantagonizing men and exacerbating the tension between the sexes. Feminists ofdifference constantly risk playing into the hands of sexist interests. In the case ofmathematics, any reference to gender differences may be interpreted or mis-interpreted as supporting the thesis of male superiority in this field. Historically,sex segregation has not always been implemented with feminist goals in mind, andseparate education has often meant inferior education for girls.

I would like to conclude this section with a few reflections on the general goalof redressing the gender imbalance in mathematics. First, there is the possibilitythat sonic feminists might not subscribe to this goal. The argument has beenadvanced that certain institutions (for example, the army or the church) are sofundamentally antithetical to women that it is preferable to stay away from them.So far, to my knowledge, no feminist has publicly advocated such an extremeposition concerning mathematics; even the most severe critics of this discipline donot advise women to turn their backs on it. However, the theoretical possibilityof rejecting nuthematics, from either a radical feminist or feminism-of-differencestandpoint, remains open.

Secondilthough this is rarely discussed, underlying most efforts to redress

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Roberta Alm

imbalance is the tacit assumption that the proper balance is roughly equal numbersof women at all education and occupational levels, up to the highest rank ofprofessorship in the most prestigious universities. Even feminists of difference,while insisting on the qualitative aspects of the balance, have not questioned thisquantitative target. To achieve this ideal proportion, in many countries, efforts toincrease the percentage of women in mathematics have focused on increasing theirnumber. However, there is another way in which the percentage ofwomen in anyfield may increase, and that is when the number of men decreases!

For groups other than feminists, increasing the number, rather than thepercentage, of women in mathematics may be a goal in itself, and not necessarilya means of achieving gender balance. Governments, for instance. may be interestedin increasing women's participation in mathematics, scieni e and technologymerely because they want to increase the total population involved in scientific andtechnological production. Likewise, mathematics departments may seek to attractmore female students as a means of remedying declining enrolments. AlthoughfemMists may take advantage of such situations, it is important to keep in mind thedistinction between the goal of increasing women's participation and the goal ofredressing gender imbalance. How would we react to the idea of the percentageof women in mathematics chmbing to 70 or 90 per cent? And, would we then behappy to see women fill positions that men no longer wanted?

Conclusion

Gender imbalance, ill mathematics as elsewhere, is a political issue. I have arguedthat strategies for redressing the imbalance rest on a variety of feminist philosophiesthat differ considerably from each other.' Making these underlying philosophiesexplicit will increase our ability to evaluate the possible advantages and dangers ofeach approach. Furthermore, it will show that we do not all share the sameattitudes and ideals, and it will prevent the frustration caused by the unrealisticexpectation that we do.- Because of the relevance of philosophical and politicalperspectives to our work, we probably will not be able to agree on what the beststrategy is for redressing gender imbalance. This is not peculiar to our field. Tochoose another example close to our experience, mathematics teaching too isshaped by diverse philosophies of mathematics and education, even though theyniav remain subconscious, and for this reason it is unlikely that we will ever reacha consensus on what is the best way to teach mathematics. However, being awareof our often implicit perspectives will contribute to a better understanding of themotivations, aims and effects of the various optionsind the extent to which theyare consonant with our beliefs and values.

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Feminism and Strategies .fiv Redressing Gender Imbalance in Mathematks

Notes

1 Several other frameworks have been employed for classifying feminist trends. Somerely on political or philosophical schools of thought or social movements that existindependently of feminism (for example. black, liberal, socialist, Marxist, existentialist.structuralist, post-modernist). Offen (1988) distinguishes between individualist andrelational feminism, others refer to successive waves or generations of feminism (notnecessarily defined by chronology). I have chosen Descarries-Belanger and Roy'stypology because it is more intrinsic to feminist thought and leis open to valuejudgments.The word 'radical' is used here, in the sense of 'root', to describe someone who wantsto go to the root of-a problem, to concentrate on its most fundamental asrcts.

3 A similar standpoint is taken by Spender (1982). Spender shows how in mixed classes,boys learn to dominate girls and girls to submit. For girls, she argues, mixed-sexeducation turns into indoctrination and practice in the art of subordination (Ibid.,pp. 118-20). See also Chapter 23, this volume.

4 I have described at length elsewhere both the project of a finninist critique ofmathematics and science (Mura, 1991, 1987) and mathematicians' reactions to it(Mura. 1992-1993).

5 Hanna (1994) articulates this kind of criticism.6 Research on gender differences, of course, is another matter. It too must rest on some

philosophy of gender relations, but not necessarily on a feminist one.7 Lee (1992) provides a good illustration of contrasting attitudes towards gender issues

in mathematics.

References

BI I I NI:v. M.F., Ca B.M.. Gotion u. N.R and TAR.1.11., j.M. (1986) ll'ottien's fi'ays(f "The Development of Self, l'oice, and Mind, New York, Basic Books.

BUM( )N, L. (1987) 'Women and mathematics: Is there an intersection?', ImernationalOrganization of II 'men and Mathematics Education N'ewsletter, 3, 1. pp. 4-7.

Di ',CARR II sr-B0 ANGI lc, F. and R(sy, S. (1991) The II mien 's Movement and its Currents ofThought: A Thrological Essay. The CRIAW Papers. 26. Ottawa. Canadian ResearchInstitute for the Advancement of Women.

FRANK1 xs I I IN, M. (1983) 'Teaching radical math: Taking the numb out of numbers'.Science jOr the People. 15, 1. pp. 2-17.

Fit ANKI xcii IN, M. (1991) 'Criticalmathematics education: Towards a definition', inQt ;11 Y, M. (Ed) Pnyeedings if the 1991 Annual Aleetitig of the Canadian lathematicsEducation Study Group, St. John's, Memorial University of NewfoundLind,pp. 109-21.

FRI tici . P. (1971) Pedagogy of the Oppressed, New York, Seaview.HANNA, G. (1994) 'Should girls and boys be taught ditferently?'. in Bn iuii it, R.. S i 1( (1/,

S IRA \NI K. R. and WINK1 I MANN, B. (lids) Didwtics of:Ilathematic., as ,IScientyicDisaphne, Dordrecht, Kluwer Academic Publishers, pp. 303-14.

KAN It -MI X.11 K, G. and R 0.1 Rs, P. (1994) 'Gender nd mathematics education', IIIGALII IN, C.. 1-1( ((SON, B.R.. W11111111, ).H. and Ea MARI (Eds) Proovdings ofthe 7th International Congress on .11athematical Education, Sainte-Foy, Les Presses de

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l'Université Laval, pp. 304-9.LEL, L. (1992) 'Gender fictions', For the Learning of Mathematics, 12, 1, pp. 28-37.MURA, R. (1987) 'Feminist views of mathematics'-Issodation.for llImien in fathematies

Newshver. 17, 4, pp. 5-10. Reprinted in International Organiz-ation of andMathematics Education Newsletter, 3, 2, pp. 6-11.

MuRA, R. (1991) Searching for Subjectivity in the 11;Irld of the Sdences: Feminit l'ieuromts,The CRIAW Papers. 25, Ottawa, Canadian Research Institute for the Advancementof Women.

M.Lasn, R. (1992-1993) 'Les critiques feministes de la science: Une menace aux femmeset la science? Analyse de deux reactions du milieu mathematique'..4tlantis, 18, 1 and2, pp. 3-24.

OH+N, K. (1988) 'Defining feminism: A comparative historical approach'. Signs:Journal of11Imen in Cidture and Society, 14, 1, pp. 119-57.

Rot ;ilk\ P. (1990) 'Thoughts on power and pedagog', in BUR I ON, L. (Ed) Gender andMathenhuics: An International Perspective, London, Cassell, pp. 38-46.

Noi D. (1982) hwisibk Ilinnen: The School* Scandal, London, Writers and Readers.TomAs, S. (1978) Overcoming Math Anxiety, New York, W.W Norton and Co.

162

Women's Ways of Knowing inMathematics

Joanne Rossi Becker

171a roars whidi lies MI the other side of silence.(George Eliot, 1871-2)

Introduction

In this chapter. I discuss the ideas in a book that I think has major implications forhow we can encourage girls and women to pursue mathematics and mathematics-related careers. ' II 'otnen's I Iiys (y- Knowing: The Development (f Self; and Mind.by Mary Field Belenky, Blythe McViker Clinchy, Nancy Rule Goldberger, and JillMattuck Tarule (1986), explores how women come to know. After brieflysummarizing the major ideas in the book. I will discuss implications for theteaching of mathematics. and raise issues which I think would benefit from furtherresearch.

Background

Feminist theory is usually developed in one of two ways. Individuals, eitheraddress, in the initial stages of their work, how a particular issue affects or is affectedby women and their concerns, or they re-examine the work of others to see if theirconclusions fit with women's experiences. All too often, conclusions based onwork with male subjects are presented as though they are indicative of all persons.

It is the second approach to developing feminist theory that has received themost attention. These re-examinations are necessary becatise so much research hasbeen based on all-male samples and thus the theories generated often result inwomen beconfing a disadvantaged group. Probably the best-known effort tocritique a widely accepted theory and assess its validity for women is CarolGilligan's work (Gilligan, 1982). Gilligan exannned Kohlberg's stages of moraldevelopment, which he based on an all-male sampleind asked whether the

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hierarchical stages he proposed were truly reflective of the way in which moraldevelopment occurs in women. She found they were not. In terms of Kohlberg'shierarchy, women seldom reach the highest levels of moral development. Gilligan,however, found that moral development in women followed a different path andwas based on different values. In a profound way, Gilligan made the case thatwomen were speaking in 'a different voice'. It was not that the values and worldview of women were better or worse than those of men, it was just that theirsystems were different. Gilligan called for an honouring of both value-systems, as

well as for both voices to be heard, thereby freeing everyone to look at how andwhat they value, and decide which way is right for thern individually.

It has become acceptable for women to proclaim their 'different voice' withregard to moral issues. It is far more dangerous, however, to broach the hypothesisthat women have a 'different voice' with regard to the cognitive. The questions are:Is the way we, all of us, are supposed to know things also determined by how somemales came to know things? Is there a male model for knowing that, in its veryformation, excluded women, denied their truths, and made them doubt theirintellectual competence? In Women's Inys qf Knowing. Belenky and her colleaguesexplore the ways in which women know, and also how these ways of knowingdiffer from those of men. In particular, they present a re-examination of WilliamPerry's model of intellectual development (Perry, 1970).

Let me address the reservation that some of you may have about describinga particular way of thinking as 'women's way'. When I refer to women or girls inthis chapter, as when the book refers to women in Women's Ways of Knowing, I amnot automatically referring to all females. Here I am generalizing, not asmathematicians do, meaning every woman, but as social scientists do, meaningmost women. The word `women' is used to refer to all those individuals whothink, come to know, or react in a fashion that is common to the majority ofwomen. These individuals may be females or males. Also, the use of the word'women' to describe the way some people think does not preclude the possibilitythat some women do not think in this way.

There is, of course, the danger that acknowledging women's different ways ofknowing will serve to reinforce stereotypes that demean women's capabilities (seealso Mura, Chapter 19 this volume). We have to make the case that 'different' doesnot mean one way of thinking is better than another. On the other hand, researchcan help provide evidence to support or refute the possibility of a women's wayof knowing in mathematics and, if it is supported, demonstrate how this can helpus understand and improve women's participation in mathematics.

Women's Ways of Knowing

The authors of 11 mmi II iy of Knowing spent years interviewing 135 women ofvariou, ages and socio-cultural backgrounds. These women were African-American, European American, and Hispanic, and were identified in formal sitesof education such as prestigious women's colleges. coeducational colleges, urban

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flomen's It'ays qf Knowing in Ilathentatics

Table 20 1: Stages of Knowing

Stage of KnowingC>

SilenceAccepts authority's verdict as to what istrue.

Received knowingLearns by listening; returns words ofauthority Speaker is not source ofknowledge.

Subjective knowingInner voice says 'I only know what I feel inmy gut Assumes there are right answers

Male version: 'I have a right to my opinion.'

Female version: 'It is lust my opinion

Procedural knowingVoice of reason; begins to evaluate validityof argument

Separate knowingLooks to propositional logic impersonalway of knowing

Connected knowingLooks to what circumstances lead toperception; wants access to otherpeople's knowledge.

Constructed knowingEffort to Integrate what is knownintuitively and what other people knowAppreciates complexity of knowledge.

Source Belenky et al . 1986

Statement

An inner voice expresses awareness thatteachers think base angles are equal

'I know that base angles are equal becausemy teacher says so.'

'I know that base angles are equal. Just lookat them, they're equal

'I know these are equal, but maybe all baseangles are not. I need a proof.'

'I know that it looks that way, but? What aboutthe triangles that other people looked at?Let's look at those too:

. _ .

'Let's physically compare the angles.'

'Tell me why you think that base angles areequal

community collegesind high-schools as well as in sites of informal education suchas human-service agencies designed to support women in parenting. Based onhours of interviews with these informants, the authors propose 'stages' in knowingthat differ in some fundamental ways from how men know. In doing so, theanthors do not perceive these stages in the same way that we understand Piaget'sstages. They are not necessarily meant as a developmental sequence through whichall learners pass. These ways of knowing do. however, iepresent.a progression fromdependence to autonomy, from uncritical to critical. As with all theories, completeunderstanding of these stages will take time. In Table 20.1, I have attempted toillustrate the stages using the statement 'The base angles of an isosceles triangle areequal' and indicating what an individual might say in each stage. These statementsgive a glimpse of what is going on for the knower.

In the Silence knowing stage, knowing is subliminal; it does not belong tothe individual and is usually not vocalized. All sources of knowing are external and

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come from authorities. The knower does not believe that she can learn from herown experience, and merely accepts or relies on an authority for all knowledge,which she does not question. Here, the inner voice of the knower would expressan awareness that teachers think base angles are equal.

In the Received knowing stage, people learn by listening. Knowing comesfrom what authorities say, and the student depends on authorities to hand downthe truth. Thus, knowledge is dependent upon an external source. There is nosense that the individual can create her own truths. Here, the knower would say,'I know that base angles are equal because my teacher says so.' I frequentlyencounter received knowers among my college students. These are the individualswho, when asked why you cannot divide by zero, tell you the reason is 'my teachertold me'. They return the words of an authority. Learners in this category do notask why a given rule is so, or wonder who gave their teacher the power to makesuch a decision. The authority that conies with being a teacher is all that is requiredfor such students to accept the truth of a statement.

The Subjective knowing stage is a powerful one for the knower andlegitimizes women's intuition. Here, knowledge derives from within, from thatwhich feels right. Knowledge no longer conies from outside the knower, and aninner voice lets the individual know that she is on the right track. In this stage, theknower would say, 'I know that base angles are equal. Just look at them; they'reequal.' Men and women handle this type of knowing differently. The male versionasserts, 'I have a right to my opinion,' or 'It is obvious.' For women, there is aconcern that their views do not intrude on those with opposing views. Theirversion is expressed as, It is just my opinion,' or 'I guess I feel so.'

For Procedural knowing to occur, a person usually requires some formalinstruction or at least the presence of knowledgeable people who may serve asinformal tutors and are able to model the process of providing evidence in supportof an idea. At the procedural stage, the methodology used in providing evidenceand presenting knowledge becomes important. In some instances, a particularmethodology may be upheld as the way.

Two types of procedural knowing are identified in the book. Separateknowing is based on the use of impersonal procedu p--. to establish truths. It isparticularly suspicious of ideas that 'feel right' (as in ojective knowing). It oftentakes an adversarial form which is particularly diffic for girls and women, andseparate knowers often employ rhetoric as if playing a game. The goal of separateknowing is to be absolutely certain of what is true. It is better to eliminate apossible truth than to accept as true something which later may prove to be Eike.The 'separate knower' would say, 'I know these are equal, but maybe all base anglesare not. I need proof: After this, she would go on to follow the rules of discoursein order to prove the statement.

Connected knowing, the second type o: procedural knowing, builds onpersonal experiences. It explores what actions and thoughts lead to the perceptionthat something is known. Experiences are a major vehicle through which knowingsomething takes place. Authority derives from shared experiences, not from poweror status. A creative process would he used to gain experiences from which a

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Table 20.2: Procadural Knowing

Separate Knowing Connected Knowing

Logic IntuitionRigour CreativityAbstraction HypothesizingRationality ConjectureAxiomatics ExperienceCertainty RelativismDeduction InductionCompleteness IncompletenessAbsolute truth Personal process tied toPower and control cultural environmentAlgorithmic approach ContextualStructure and formality

Source: Gilligan (1982)

conclusion could be drawn. The 'connected knower' would say, 'I know that itlooks that way, but? What about the triangles that other people looked at? Let'slook at those too.' In answering the question, 'Why do you think that?', the'separate knower' would look to deductive logic. The 'connected knower', on theother hand, would want to know what circumstances led you to that conclusion.

It is in this stage, procedural knowing, that there is the most conflict with thetraditional way of knowing in mathematics. If the only knowledge accepted asvalid is that which can be statistically demonstrated or is based on deductive logic,methods that are independent of the knower's actions, then that which is knownthrough induction would be devalued. And in mathematics, is not knowledgedeveloped through deduction viewed as more valuable than knowledge gainedthrough induction? But, how do we know what to set out to prove (separateknowing) if we do not first know things through inductive reasoning (connectedknowing)? Let us look at some words we associatc with these two kinds ofknowing (Table 20.2). React to them as valid processes in knowing mathematics.If it is true that women tend to be 'connected knowers' and men 'separateknowers', and taking into account the way we value each of these ways of knowingand the way we teach mathematics, might this help account for the relatively smallnumber of women pursuing mathematics-related fields? The authors have no 'harddata', to use a separate knowing term, to support the notion that women favourconnected knowing, but such a preference would certainly fit into Gilligan's(1982) categorization of the conceptions of self. In fact, the authors of Mmen'sI3/4ys of Knowim have borrowed the words 'separate' and 'connected' fromGilligan.

The last stage in the model is Constructed knowing. Here, as the nameimplies, all knowledge is constructed by the knower. Answers are dependent onthe context in which the questions are asked, and on the frame of reference of theasker. This type of knowing is particularly relevant to mathematics; for example,the solution of an equation is dependent on the domain being considered, and

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what can be proven is dependent on the axioms being assumed. It is in this stagethat the learner can integrate her rational and emotive thoughts. as well asappreciate the complexity of knowledge formed from various perspectives. Here,the knower would use the creative aspect, induction, together with the rules ofdiscourse, deduction, in order to know something. The constructed knowerwould say, 'Let's physically compare the angles. Tell me why you think that baseangles ...' There is a willingness to describe how the knowing occurred. Inconstructed knowing, the artificial dichotomy between separate and connectedknowing, presented in Table 20.2, becomes apparent and the two ways of knowingcan be merged. as recommended by Moody (1989) and others who think andwrite about feminist science. Certainly this category of knowing fits very well withcurrent constructivist views on the teaching and learning of mathematics (Davis

et iL. 1990).

Teaching and Learning Mathematics

Having outlined the stages proposed in Iliwren's 11i-zys of Ktwwing. the issue I wouldnow like to explore is what the implications of this theory are for the teaching ofmathematics. The authors make a plea for what they have termed 'connectedteaching'. Connected teaching would address the issues important to the learningof women identified in the book, and thus might help students develop intoconstructed knowers. Some ideas for connected teaching are listed in Table 20.3below. w'aich I have adapted from Buerk (1985) and Morrow (1987).

hri connected mathematics teaching, one would share the process of solvingproblems with students, not just the tinished product or proof. Students need tosee all the crumpled papers we put in the wastepaper basket, if they are tounderstand that matheniaticians do not arrive at a solution the first time or the firstway This means that when we start teaching from a new mathematics book, weshould not solve all the problems before class. Instead, we could show students howwe start ,1 given problem, make an error, and begin our solution over again. Femalestudents need to watch women professors solve (and fail to solve) problems, andmen professors fail to solve (and solve) problems. They need models of thinkingthat are human, imperfect, and most of allittainable.

Mathematics needs to be taught as a process, not as a universal truth handeddown by some disembodied. non-human force. Mathematics knowledge is not apredetermined entity. It is created anew for each of us, and all students shouldexperience this act of creation. Presenting mathematics in the 'commercial with(male) voice-over' mode, as disembodied knowledge that cannot he questioned,works against connected knowing. The imitation model of teaching, in which thennpeccable reasoning of the professor as to 'how a proof should be done is

presented to students for them to mimic, is not a particularly effective means oflearning for women.

In connected teaching, teacher and students would engage in the process ofthinking and discovering mathematics together. Alternate methods of solution

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Women's Ways of. Knowing in MatIumatics

Table 20.3. Examples of connected teaching in mathematics

Issue

Voice

First-handexperience

Confirmation ofself as knower

Problem-posing

Believingversusdoubting

Support versuschallenge

Structureversus freedom

Importance for connectedteaching

Education occurring in thecontext of conversationGaining a sense of selfDeveloping one's ownauthority

Building on intuitiveunderstandingValidating students'knowledge baseProviding insight into reasonsfor area of studyEncouraging activity versuspassivity

Providing a basis for area ofstudyBecoming a rule makerBecoming a constructor ofknowledgeProviding a basis for midwifemodel of education

Focusing on process ratherthan outcomesAllowing students to seeuncertainties and understandknowledge construction

Giving students alternativemodes for discourse

Allowing students to becomeindependent learners

Giving guidance/ment-vingwithout imposing tyrannicalexpectations

Source Adapted from Buerk (1985) and Morrow CI 987)

Connecting teaching inmathematics

Students develop ownjustificationGroup problem-solvingWriting exercisesConfidence-buildingworkshopsClass discussionsStudent-designed projectsTeacher does not giveanswers

Applications-orientedworkshopsDrawing and constructingmodelsUsing visual representationsUsing computers

Looking for what is alreadyknown

4 Listening to students' reasonsRespecting students' ideasEngaging class in outsideactivities

Allowing students to strugglefor solutionHaving high expectationsFocusing on explanationSharing our problem-solving

Asking for right explanationseven when rightAnswering questions withquestions

Validating presentunderstanding while providingchallenges

Following prepared curriculumwhile allowing for explorations

would be encouraged. In problem-solving, for example, more emphasis would beplaced on finding ,mother way to solve a particular problem, than on solvingadditional problems in the same manner. Students would use their intuition in an

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Joanne Rossi Becker

inductive process of discovery. Here, the emphasis would be on generatinghypotheses. For example, many polygons could be examined to determine thesum of their interior angles and in this way generate a formula for the sum of thcinterior angles of any polygon. Patterns could be used to discover why the productof two negative integers is a positive integer. The generation of these relationshipswould be valued as much as their verification, for without this prior inductivework, we would be at a loss to know what to prove.

A metaphor used by the book to describe connected teaching sees the teacheras a midwife, nurturing the student's newborn thought in a supportive environ-ment. The first concern of this midwife/teacher is to preserve the student's fragile,newborn thought. The midwife/teacher then supports the evolution or growth ofthis thought. 'The goal is not to replace it with a different, teacher-generated,'better' thought. Rather, it is to help the student's thought grow, mature, anddevelop. Both the teacher and the student engage in this process. Their rolesmerge, and the teacher of students and the students of the teacher cease to existwhile a new team emerges: teacherstudent and studentsteachers (Freire, 1971).

In connected teaching, groups are created in which members can nurture eachother's thoughts to maturity. Here, no one needs to apologize for uricertainty.Questions are posed and potential answers are explored together. More courseswould be conducted in a style of community, and fewer would be taught in themasculine style of adversarial discourse or hierarchical pontification. Truth orknowledge is constructed through consensus, not through conflict. Diversity ofapproach is welcomed in discussions. Connected teachers trust ;.heir students'thinking and encourage them to expand upon it. This is particularly importantbecause the women reported in the Belenky et al. (1986) study experienced doubtas debilitating rather than energizing.

Recent research provides evidence that females prefer and do better inmathematics classes which use cooperative learning in small groups, rather thanindividualized or competitive learning strategies (Fennema and Leder, 1990).Moreover, as discussed above, a collaborative approach seems to offer morepossibility for connected teaching. Certainly, a small-group environment providesan atmosphere in which different points of view can be shared and students candiscuss mathematics. And it is difficult to envision how students can construct theirown knowledge by weaving together objective and subjective knowing in amathematics class taught in a traditional teacher-centred lecture format.

There is considerable evidence that mathematics teaching at all levels neitherexemplifies connected teaching, nor encourages constructed knowing (NationalResearch Council, 1989; Weiss, 1989). How much difference might such a changein the teaching of mathematics make in the improving attitudes and performancein the subject for more students of both sexes? Some beginning work in thisdirection is positive (Buerk, 1985: Damarin, 1990; Rogers, Chapter 21, thisvolume). But, I suggest that much more research is needed to help us identify howbest to encourage connected teachingind to dm rmine what impact it mighthave, especially on women's participation in mathematics-based careers.

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Women's W'iys of Knowing in Mathematics

Questions for Future Research

In this section. I will suggest some avenues for future investigation that revolvearound Mmen's Ways of Knowing. This is a fertile area to study for it might takeus beyond the research of the last fifteen years on gender differences inmathematics education by providing a different theoretical model with moreexplanatory power.

First, I have reanalysed data collected before the appearance of Illomen's Waysof Knowing (Becker, 1990, 1984). In this research, I conducted in-depth interviewswith male and female graduate students in the mathematical sciences in order toidentifi; factors associated with their decision to attain a graduate degree. Thus. Iwas focusing on the point between undergraduate and graduate study inmathematics at which we lose many women (see Friedman, Chapter 5, thisvolume). There were no questions in my protocol which specifically adchessedaspects of the Women's Wtys of Knowing model. However, one particular portionof the interviews focused on the period at which students' interests in mathematicsdeveloped, as well as on what students liked about the subject. Although there'were no gender differences apparent in these categories, several factors whichrelate to Women's Wiys of Knowing are worth mentioning here. Beyond likingmathematics because they were gooi' at it, graduate students were attracted by theproblem-solving aspect of the subject, and particularly liked starting with certainassumptions and solving puzzles logically The objective nature of the discipline,as described by the informantstho appealed to them. They liked being able totell if a problem was solved or if a proof was correct. This observation leads to atleast one area for further research. Are women in mathematics more likely thannon-mathematicians to be separate knowers and thus be attracted to the subjectbecauseit least at this early stage, they perceive mathematics to be an objectivediscipline in which they can find absolute truth? Do their views of mathematicsevolve as they pursue further study and actually do research themselves?

A second factor I observed in my studies was that nearly all informantsdeveloped their liking of mathematics quite early and before high-school.Frequently, a teacher was mentioned as the one who piqued their interest byproviding an enriched curriculum which went beyond arithmetic to includeproblem-solving or topics in algebra. Thus, there does seem to be the possibilitythat teachers and methods of instruction might make a difference in students'ultimate career choices. Could an extensive use of connected teaching affect morestudents in this positive way?

Other questions which arise from the model as described in this chapterinclude the following:

Does the model represent how women (and men) conic to knowmathematics? How might we proceed in testing this model in mathe-matics?Do more men fall into the separate knowing category and more womeninto the connected knowing category? Is there any relationship between

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these categories and career choices? How might one begin to ascertain thetype of procedural knowing category into which an individual fits?Can the model help to explain the continued underrepresentation ofwomen in mathematics-based fields?

Because I believe that Wnnen's Ways of Knowing has important implications forthe teaching of mathematics, additional questions relating to this area arise. Thisbook does provide evidence that the traditional style of teaching mathematics is inconflict with how many women learn. Connected teaching, which emphasizesconnection over separation, understanding and acceptance over assessment anddoubting. and collaboration over debate, may be a more effective technique. Weneed to investigate questions such as these:

Can a change to connected teaching at all levels reclaim mathematics andscience for women students who have already been turned off the subjects?For example, instead of giving up on women already in college who arechoosing fields which do not require mathematics, can we attract them tomathematics and science by good teaching in entry level courses? Such astrategy has been suggested by Tobias (1990).Does connected teaching make a dii rence in student performance and inattitudes towards mathematics?How can we help mathematics teachers at all levels become constructedknowers and connected teachers?

Conclusion

As mathematics educators continue to investigate ways to increase women'sparticipation in mathematics and mathematics-related careers, the ideas in

11;lys of Knowing should be given considerable weight. In so doing, themathematics education of all students nnght be enhanced by the wider applicationof connected teaching. If the way mathematics is currently taught alienates manywomen because it does not appreciate or validate their ways of knowing, thenmany women may choose not to pursue mathematics and mathematics-relatedcareers.

What are the political implications of broahing the possibility of a differentway of knowing? Should we fear that sonic will use this difference as a means ofproving women's inferiority? There is always a danger that knowledge can be usedagainst people, but there is a greater danger in not addressing this issue at all. Ifthere is the chance that connected teaching could enhance the education ofwonwn, then it is an approach that should be seriously considered. If movementfrom subjective to connected knowing is based on experience, and if connectedknowing is women's preference, then, and assuming we waot them to know, wemust present information to women in a different way and accept another way ofknowing. To ignore these differences in how women and men conic to know is

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Women's Ways cg-Knowntg m AIatlulnancs

to deny large numbers of people access to mathematical knowledge. Addressingthe implications of Ifinnen's Wiys of- Knowing will not only benefit female students,but society at large. Who can say what that roar which lies on the other side ofsilence might accomplish in mathematics?

Note

Many of the ideas in this chapter have been developed through discussion with JudithE. Jacobs (see, (or example, Jacobs (1994)). The author acknowledges her inspirationto pursue this topic, her scholarly thoughtfulncssind hei collegial sharing of ideas.Portions of this chapter haw been published by Jacobs and Becker in the newsletterof Women and Mathematics Education and an earlier version was discussed at aSymposium of IOWME in June 1991.

References

Bi iNtR. J.R. (1984) 'The pursuit of graduate education in mathematics: Factors thatinfluence women and men'. .fournal o Educational Equity and lAwdership. 4. 1.

pp. 39-53.K I R. J.R. (1990) 'Graduate education in the mathematical sciences: Factors influencingwomen and men', in Bt iti oN. L. (Ed) Gender and Mathematics: An InternationalPaspective, London. Cassell. pp. 119-30.NKY, M.E. Ci !MIN B M.. Goi I Mil N.R. and TAKI..1 1 J.M. (1986) filmic', 'sII;iys of Know*: 7he Development of Self, ii)ice, and Mind. New York. Basic BooksInc.

Br'l IkK, a (1985) 'The voices of women making meaning in mathematics'. Journal ofEducation, 167. 3, pp. 59-70.

DAMAILIN. S. (1990) 'Teaching mathematics: A feminist perspective', in Ci )Nt T.J. andHilt M11. C.R. (Eds) li.achiv and Learning Mathematics in the 1990s: 1990 Ii.arbook,Reston. VA. NCTM. pp. 144-51.

Dwis. R.B.. MAI n R. C.A. and Ni N. (1990) Construanist !len', on the 'Kuchingand Learn* of Matlwmaties, Reston. VA. NCTM.

El 1(11. G. (187; -2) Middlernarch. New York, Penguin Books.FENN1 MA, E. and LH)! It, G. (Eds) (1990) Mathematics and Gender, New York, Teachers

College Press.FRI Ilk . P. (1971) Pekogy (y. the Oppressed, New York, Seaview.G1111(,,AN, C. (1982) In a Different l'oice: Psychohn "lhecny and II iw;c1C., Development,

Cambridge MA, Harvard University Press._Inc( nus. J.E. (1994) 'Feminist pedagogy and mathenutics'. ZAmralblatt dcr

Mathernatik, 26, 1, pp. 12-17.Mui )0Y. J.B. (1989) 'Women and scient e: 'Their critical move together into the 21st

century'. NIISAction. 2, 2. pp. 7-10.Mt not: Av. C. (November. 1987) 'Connected Teaching in M.-1,einatics'. Paper presented

at the annual meeting of Research on Women and Education, Portland, OR.I( /NA I Ri NI Alt ( 11 (.:(11'Nt II (1989) Eiwrybody Comas: A Report ty the Future of..lIathentanc., Education, Washingt DC, National Academy Press.

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Primo,. W. (1970) Forms qf Intellectual Derelvment in the College Years, New York, Holt,Rinehart, and Winston.

TOMAS. S. (1990) They're Not Dumb, They're Different: Stalking the Second Tier, Tucson, AZ.The Research Corporation.

Wurss, I. (1989) kience and Mathematics Briefing Book, Chapel Hi 11, NC, HorizonResearch.

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Chapter 21

Putting Theory into Practice

Pat Rogers

Listen to a woman groping for language in which to express what is onher mind, sensing that the terms of academic discourse are not herlanguage. trying to cut down her thought to the dimensions of a discoursenot intended for her. (Rich, 1979)

Introduction

A forcefid impetus for changing my teaching has been reflecting on my ownexperiences in learning undergraduate mathematics and doing mathematicalresearch. When I first started teaching, I adopted those practices I had observedas a student: I lectured. I believed that teaching at the post-secondary level involvedthe transmission of knowledge from me, the expert, to the students, the novices. I

saw my job as exposing the students to the content of the course. The students'job was to master this material by listening attentively as I explained ideas, bywatching carefully as I showed them how to solve problemsind by practisingsolving problems on their own at home. I administered tests in order to measureachievement and to rank students in relation to their peers. Each lecture had anatural pattern: I introduced a topic. covered the blackboard with formulas andmathematical language. and worked a couple of illustrative examples. I asked a fewquestions. I even elicited a few answers usually from the same three or four(male) st.idents. Filially, I assigned homework. I was consideied a successfulteacher. My course evaluations proved it. Students praised me for my enthusiasm,my organization, the clarity of my exposition ny knowledge of the material andmy accessibility. The most critical comment was a request to slow down a little.Yet, when the final examination came, many students failed or wrote suchincomprehensible answers that I wondered if we had all been involved in the samecourse. How could they do so hadly, I wondered, when I had explained thematerial so well? How could they r;ot buv the goods I had sold them sopersuasively? Of course this trouble,1 me, but it wa, easy enough to dismiss:'Students wine to university so ill-prepared. If only we had better students. just

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Pat Rogers

think what We could do!' A familiar refrain?Over the years I became increasingly concerned about the students in my class

who were silent. These were students who sorely needed individual attention yetnever used my office hours. These were the students who were heading for certainfailure but were lulled into thinking they might pass the course because, as theyoften said, 'It seems so easy when you do it on the board The majority of thesestudents were female. I was able to experience their private distress only when Ibegan to reflect on my own experiences as a mathematics undergraduate atOxford. lt shocked me to realize how faithfully I was reproducing in my ownclassroom the structures which had so effectively silenced and Liisempowered methen.

It seems to me there are two important questions to consider when examiningthe ways in which traditional mathematics pedagogy at the post-secondary levelmay disempower students. Whose mathematics are we teaching? How are weteaching it?

Whose Mathematics Are We Teaching?

As a high-school student. I was able to reconstruct for myself any formulas, rulesor results that I needed. Some of my teachers even admitted to not knowinganswers to many of the problems that I posed. However, at the university theprofessor was distant, remote and expert and I began to experience mathematicsas something distant and imposed, not something that I had a role in producing.We were rarely given the opportunity to play with mathematical ideas or toconstruct our own meanings (except on our own at home). Instead, through themedium of the polished lecture (or textbook), mathematics came to me finished,absolute and pre-digested.

A pedagogy that emphasizes 'product deprives students of the expc.1-ience ofthe 'process' by which ideas in mathematics come to be. It perpetuates a view ofmathematics in which right answers are the exclusive and sole property of experts.Such a pedagogy strips mathematics of the context in which it was created andreinforces misconceptions about its very nature. Students permitted to see only thepolished product may come to believe they can never create sinnlar results forthemselves ('I didn't know where to begin!').

Cirol Gilhgm's (1982) research in the field of women's moral developmentprovides a hulk between 'math avoidance' and mathematics pedagogy.= Sheidentified two styles of reasoning, which although not gender-specific arc thoughtto be gender-related: one, the traditional style, is characterized by objectivity,reason, logic. and an appeal to justice; and the other, 'the different voice', oftenidentified with women and as a consequence devalued, is characterized bysubjectivity, intuitionind a desire to maintain relationships. The 'different voice'is rooted in an ethics of care and responsibility.

Dorothy linerk (1985) has studied Gilligan's styles di relation to the work ofmathematicians. She presented a group of mathematicians with two lists of word,

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Putting 'Theory into Practice

describing examples of 'separate' reasoning (the traditional style) and 'connected'reasoning (the different voice) in the field of mathematics. They unanimouslyidentified the 'connected' list as representing the way mathematicians domathematics. IMIathematics is intuitive", they said. They stressed the creativeside: attention to the limitations and exceptions to theories, the connectionsbetween ideas, and the search for differences among theories and patterns thatappear similar. And yet they agreed that the "separate" list comyed the way thatmathematics is communicated in the classroom, in textbooks, and in theirprofessional writing' (Buerk. 1985).

In reality, mathematicians employ both forms of reasoning in their v. ork.. Butthe problem with mathematics teaching, particularly at the post-secondary levelwhere the lecture mode of instruction is so predominant, is that the creative-intuitive form is largely eliminated. Students are not given the opportunity to beinvolved in the process of constructing mathematical ideas, a process in which'connected' thought is so important. There is an enormous cognitive gulf betweenthe way mathematics is presented and the individual ways in which it is possibleand natural to arrive at an understanding of it, or to construct it for oneself. Somestudents are able to bridge this gap for themselves, but many are not.

Of course, the expository approach to teaching mathematics affects allstudents, but it does not affect them all to the same extent. If Gilligan's claim thatwomen favour a 'connected' reasoning style has validity, then the tendency of mostteachers to take only the traditional 'separate' route ,o mathematics could inhibitor prevent some students especially women students from claimingmathematics for themselves.

How Are We Teaching Mathematics?

One important component of the expository method of teaching is the use of thedistant authority to 'impart knowledge'. Practices, such as lecturing, subordinatesstudents' knowledge and understanding to that of the professor and the even moredistant authority, the textbook. Solving problems.for students does not teach themto solve problems for themselves. Instead, it disempowers students by renderingthem passive, and conveys mistaken notions such as, 'there is only one correctsolution, namely the teacher's'. It can also give students the impression that theteachers lack confidence in their ability to solve problems for themselves, that is,to think mathematically. Such implicit judgments of students abilities as well as themore explicit verbal and written judgments they receive can stifle students'emerging styles and diminish their confidence to engage. Indeed, while studentsdo need to acknowledge that they will make mistakes, they also need to know thatthey are intrinsically capable. According to the findings of Belenky ct al. (1986),this may be particularly important for women students. The unconscious oruncritical use of authority in the classroom is a powerfial disabler.

The fact that women do tend to be more silent in the classroom is welldocumented. Belenkv et a/. (1986) point to the difficulty with the development

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of 'voice' in situations dominated by male authorities. Krupnick (1985), in herstudy of gender in the classroom at Harvard College, found that women speak lessoften than male students, tend not to compete with them, and are more sensitiveto interruption (they do not pursue their thought once they are interrupted andthey are interrupted more often than male students). Inequitable access toclassroom discourse makes it very difficult for women to cultivate and make useof an original voice. If alienation from the subject results when a student's 'sense'or experience remains unexpressed and/or uncultivated, if alienation can bethought of as a muting of the self (Lewis and Simon, 1986), and if the ability toarticulate an idea signals ownership of that idea, then silence in the mathematicsclassroom spells a serious limitation in the ability to claim a subject freely and withenergy and self-confidence. I suggest that without this sense of ownership, theoptimal development of mathematical thinking skills may not take place. Creative,subjective engagement will be curtailed, and therefore those thinking skillsdependent on such engagement may remain underdeveloped.

A Mathematics Pedagogy of Possibility

I would like to outline an alternative approach to the teaching of mathematics, onewhich is rooted not in the authoritative and imposed style which distances andsilences, but in a style which encourages direct access and engagement, freecreative expression, and ownership of the subject. This alternative iMght bethought of as introducing a finninist element.

The first element of this approach has been influenced, among other things,by an ethics of caring and responsibility that is designed to enable those who havebeen silenced to speak (Simon, 1987). One must become, first and foremosti'caring teacher'. Here I use the term caring not simply as feeling of concern,attentiveness or solicitude for another person, but in the very specific sense of'caring' as in helping 'the other grow and actualize himself' (Mayerotr, 1971). Inthis view caring teachers are able to focus primarily, or at least initially, upon thestudent rather than upon the subject mattet, with the idea that the route to thesubject is not imposed from without, hut rather illuminated from within. They donot work upon their students, but with their students, looking at the subject matterfrom their perspective and at their level. Elsewhere, I have called this a student-sensitive pedagogy (Rogers, 1988), for it is grounded in the students' ownlanguage, is focused on process rather than on content, and centres on the stud, nts'individual questions and learning processes. Students who are 'cared for' in thisway are set free to pursue their own legitimate projects (Noddings. 1984).

Another element in this approach involves demystifying the doing ofmathematics. This includes such things as calling stud-nts' attention to mathe-matics as a creation of the human mind, making visible the means by whichmathematical ideas come into being and the process by which they are 'polished'f-or public consumption. It also includes encouraging students to challengeauthoritative discourse, and to feel that the gates to the mathematical community

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are open and that they have the skills they need to walk through and to operatewithin it.

Reflecting on the differences between what I was doing in my early teachingand what I came to believe needed to be done. I realized that a core difference layin the engagement of students within the classroom in purposeful, meaningfulactwity

A Feminist Mathematics Pedagogy in Practice

In this section. I describe sonic of the ways I has e put the ideas outlined above intopractice. At the beginning of a course I discuss my goals with the students andmake the goals available in written form. They HI into three categories: content-.specific goals, process goals, and conununication/social goals. The content-specific goals include the development of students' skills in constructingmathematics and writing their own proofs clearly and precisely in convincingform. Process goals include the development of independent working skills (themore highly developed the individual working skills, the less the need for ateacher). Some of these skills include reading a mathematics text with under-standing; finding, analysing and correcting mistakes; and asking mathematicalquestions. Since mathematics is an activity which depends on communicationbetween colleagues. social goals are also stressed and include helping studentscollaborate with others; communicating ideas clearly and with confidence bothorally and in writing; listening actively and offering constructive criticism; askingand responding to challenging questions. My teaching methods vary depending onthe chai..cteristics of the particular class and the questions raised by the students.Following is a brief description of some of the methods I use most frequently.

LecturingI have not completely abandoned lectures; instead I use them sparingly for threemain purposes: to introduce a new section of material; to conclude a topic anddraw everything together; to introduce a new concept and to motivate assignedreadings and problems (this is usually a shori lecture given at the end of a session).

Think-write-pair-discussThe 'think-write-pair-discuss' idea is adapted from Davidson et al. (1986) and isthe most usefill and most frequently used of my current strategies. It is also the onethat is most adaptable to student groups of all sizes. I present students with aquestion, or give them a segment of the text to read. They work independentlyat first, put their thoughts and ideas down in writingind then form pairs todiscuss. This provides support for those students who are unsure of then ideas orwho have a fear that what they say iMght appear foolish in public. It has the effectof increasing and improving participation and involving all students. This mayprecede or be integrated into all the activities which follow.

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Whole-group dialogueLike a lecture, whole-group dialogue is teacher-directed. Unlike a lecture, it isstudent-centred. Dialogue usually follows assigned reading and thinkwritepairdiscuss activity, and operates by way of questioning. My questions aim to help

students express their ideas clearly and with precision. For this, good activelistening skills on my part are essential: I demand reasons for statements, andchallenge students to formulate their ideas in their own words and explain them

to each other. The latter is important because students are often unable to 'hear'their instructor, but are perfectly able to hear one another, and such listening and

telling is empowering.

Board workStudents conie to the board individually or in small groups to write up questions,problems, and solutions. We then discuss the solutions. I may assign problemsahead of time. and I may allow time in class for students to work on solutions ingroups prior to the board work. Sometimes, several groups will work on the sameproblem at the board. I am then able to demonstrate that there is no single rightapproach to solving a problem. Students learn from each other and learn both oraland written presentation skills. At first, I do not require that students remain at theboard to explain their solution and answer questions. However, as the classroomclimate becomes more supportive and students begin to encourage each other toparticipate, confidence in their ability to discuss mathematical ideas increases.

BrainstormingI give student pairs between two and five minutes to write down everything theyknow about a given topic. I then call on the pairs and all the information is puton the blackboard. Another way of gathering the information is to ask questionsof each student in turn, giving students having the right to 'pass' I call thistechnique 'round robin'. Discussion then centres around classifying and evaluatingwhat has been gathered. I use this technique most often for review or to begin aninvestigation (see below).

Problem-posingStudents generate their own questions in a particular area. These are thenexaminedind the students commit themselves to a particular conjecture. Forexample, one year an early conjecture in my group theory course was Lagrange'stheorem. Proving this theorem became the focus of our work for several weeks.This thcus made the course problem-driven, a response to the students' ownagenda and chosen avenues of inquiry. An activity of this sort usually occursnaturally, unlike an investigation which is artificially established.

InvesdgationThe class exammes patt:.rns thund in concrete examples in order to uncoveralgebraic structure. Generalizations are then made in the form of (onjectures, anda theory which proves or disproves the conjectures is developed. I have usedinvestigations to generate all standard structure theorems for finite groups.

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Small-group workGroups of two to four students work together while I circulate among them andcheck their work. This provides me with immediate feedback on studentunderstanding and enables students to influence the pace and the development ofthe course. Even with large groups of over 100 students, where I can check onlythe work of the students at the aisle, I am provided with immediate feedback.

Reading exercisesThe development of good reading skills is essential to gaining independence inmathematics. An effective way to help students delop good reading skills is toinsert question marks at crucial junctures in a proof before assigning it to studentsfor reading. I ask students to rewrite the proof, repLcing each question mark withan explanation. I vary this activity after the students have gained some skill byhaving them insert their own question marks. To give students more practice in'talking mathematics' I may ask them to give the explanations for the questionmarks orally either to the whole group, or by having them pair up as in 'think-write-pair-discuss' this is a particularly useful way to help students develop skillin understanding definitions. When students have more confidence I have thempair up to pull apart the statement of a theorem and write it in their own wordsprior to proving the result. All technical words must be defined. The structure ofthe statement is examined and a proof strategy outlined. This is often done inconjunction with the 'forward-backward proof' technique (see below).

Proof generationThe 'forward-backward proof technique described by Solow (1982) in his book.How to Read and Do Proq-s, is a very empowering strategy. It takes its name fromthe way mathematicians typically organize their thoughts when attompting toprove a conjecture. Although it cannot be successfully applied in all situations, ithas virtually elinnnated compLints from students in my courses that they 'don'tknow how to begin!'

Community

A classroom climate characterized by safety and trust is essential for risk-taking tooccur and for students to be willing to test their ideas and thoughts and developfluency in mathematical language. In other words. I work to build a communityof mathematicians.

Some students have difficulty adjusting to my approach to teaching becauseit includes them constantly and individually. For this reay.in, constant monitoringof my teaching is important. I do this primarily through the use of 'one-minutepapers' (Wilson, 1986). The one-nnnute paper is a simple device for obtainingimmediate feedback on the effectiveness of teaching. In one version, I ask studentsto take out a piece of paper in the List minute or tsso of a session and write downone or two points they are still confused about. They hand these in anonymouslyand I use their responses to plan the next session. Alternatively, I may ask thestudents how they tel about various aspects of my teaching. For example, through

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this device I learned that one student found my questioning techniques andpractice of asking her to come to the board were intimidating and humiliating forher. She did not like being singled out, fearing the public display of her perceivedignorance. On the other hand, she admitted that she appreciated the opportunityof learning from other students. Through negotiation, we were able to arrive ata compromise which made her, and many other students in the class, morecomfortable about participating. The result was that I began to use the 'round-robin' technique (in which students are allowed to 'pass') more frequently and havestudents come to the board in pairs. In mathematics, once a climate of trust hasbeen fjr' y established, these one-minute papers can be used very informallyduring the session. For example, I might ask students to work through an exercisewhich will demonstrate whether they have understood a concept we have justexplored. While they are writing, I walk around and check their work to see howwell they have understood the topic before proceeding to a new one.

Evaluation of Student Learning

I evaluate students' learning constantly in the sense that the student-centredteaching techniques and the one-minute papers I employ provide immediate,frequent and regular feedback on students' understanding of the course contentand processes. Following is a brief description of sonic of the more formalevaluation methods I use.

I assign homework regularly throughout the course and space these assignmentsso that, when combined with quizzes and examinations, students rec::.ive regularfeedback on three kinds of written work. I encourage students to collaborate onassignments but accept only independent write-ups. I comment on their work butdo not grade it because I want them to experiment freely with the cooccpts andprocesses of the course without being penalized for doing so.

Students can earn participation (-tea in A variety of ways, allowing tbr theirindividual learning preferences. Some of these ways include visicing use tbr anoffice consultation, presenting aspects of assigned readings in class, participating inassigned collaborative exercises, sharing ideas by coining to the blackboard andpresenting a proof, asking questions, offering explanations. or joining in discus-sions. The issue of participation credit in the final-grade calculation is a matter ofsome concern to me. I have used it often in the past but have found that it cant-oster competition, regardless of the quality of classroom climate I have been ableto establish.

I give tests regularly, but only in conjunction with giving students opportun-ities tbr improving their grades but more importantly their comprehension of thecourse material. When I first ASSL", a students work. I simply indicate where theargument breaks dowil and award a mark. with no explanation. Occasionally. I askA question calling for clarification. The student then responds to me in writing.The impottance of giving students a second hance far outweighs its value inincteasing their grade An early failure in a course can be very dispiriting. As well.I believe that lusts should and can be a vehicle for learning. and not ,,olely an

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instrument of evaluation. Many students tell me that they appreciate theopportunity to learn from analysing their own work. I have also found that itimproves the learning environment in the class and reduces students' test anxiety.

I also believe in formal examinations. Formal examinations are like finalperformances, an opportunity for students to pull the course together, to see it asa whole and to demonstrate how much they have learned. I have experimented,using cooperative learning techniques (see, for example. Aronson et al., 1978),with giving students responsibility for generating final examination questions. Theresult of this experiment in collaborative examination design was that it fostereda cooperative spirit among the students towards preparing for the examinationindeed they probably learned more from this activity than from the examination

Conclusion

My first experience in employing participatory teaching methods dates back to asecond-year finite mathematics course. Two other parallel courses were taught thatyear by otb...r instructors using traditional lecturing methods. My main objectivethen was to prove to my colleagues that I could teach sixty students withoutlecturing, without compromising standards, and without disadvantaging studentseither because I 'covered less material or because they achieved lower final gradesthan students in the two parallel courses (these are the criticisms commonly madeby mathematicians in regard to those who abandon lecturing). The result wasencouraging and convincing.

Measured by a common final examination. the three courses had almostidentical averages. However students in my course obtained more of the highestgrades osTrall and far fewer of them failed. Covering the course content pi esentedno difficulty, either for me or for the students, judging by their examinationsuccess. Over two-thirds of the students participated actively in the course in Coeway or another and only One student complained of any discomfort with mvteak. hing methods.

judged by examination criteria alone, one canhot conclude that the lecturemethod came out second best. However, it is clear that in this first try, my approachheld up well against the traditional approach. On the other hand, it is questionablewhether the impact of particidatory. democratic teaching methods can readily beevaluated in the short-term, or by quantitative means. I was in fact more interestedin discovering whether I could teach this way and whether students would beresponsive to these methods and learn from them. This took precedence in mymind over coniparing these tcclunques with cmiventional methods or trying toprove their superiority.

I return now to the project with which I began this discussion, that of,leveloping a pedagogy designed among other things to suit women's styles oflearning. Since that first experiment I have further refined my approach toparticipatory teaching in both dvanced mathematics courses and in courses foi

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Pat Rogers

pre-service teacher candidates. Whatever the level of the students, whatever theirbackground preparation in mathematics, however advanced or elementary themathematics included in the course, the results have been the same. My coursesenjoy high demand, lower than usual levels of attrition and absenteeismindhigher and more consistent achievement. What is more important to me, however,is the satisfaction students exhibit as they become fluent and competent with theprocess of doing mathematics. A consequence of teaching in this way, aconsequence I have found more rewarding and compelling than any other, is thestrong sense of community and caring that develops among students. Students aremore concerned about each other's welfiire and progress than anxious to competewith each other for higher grades. Students develop close bonds of loyalty to oneanother which carry well beyond the course itself and are often long-lasting. Lackof competition is not usually associated with the mathematics classroom (however,see Rogers (1988)), yet it appears to be a very natural outcome of my teaching now,and evidently provides a more comfortable and equitable classroom environmentfor the fenule students I teach.

Now that I know I 6711 teach this way. I discover that I prefer teaching this way.It allows me to know the students well and I am now much more aware how theycomprehend the material I do not have to wait until the final examination tofind that out, as is so often the case when one lectures. By providing opportunitiesfor students to hear and to develop their own voices through engagement inauthentic mathematical activity within the classroom. I am able to engage themin yurposeful, meaningful] academic discourse, allowing them to claim ownershipof mathematics for themselves. In so doing. I believe I not only avoiddiscriminating against students who are currently denied iccess to mathematics(especially women). but I also provide a more meat ingfill and equal mathematicseducation for all students.

Notes

I This chapter is a revised version of ROget (1992).2 I use the term '111.ith avoidance because it is well known. However, my own view is

that society has no,eti to exclude women from the study of mathematics. Labellingwomen's absence from mathematics 'avoidance' is a monocultural view because it failsto recognize that it is the discipline (in both kenses) of mathematics itself which turnsvoinen away. W(nnen may indeed choose to avoid matheinatics, but the term 'mathavoidance' does not connote a deliberate choice.

References

ARcrs.s, vsd V, N., S ill IAN, C., Sni N J. and SN,' Po M. (1978) The jipau,Be\erley I lills. CA, Sage.

NKr, M.F., Ci IN( '1 B.M., iI Hit sit is. N.R. and TARI'l 1 , J. M. (1986) II .011/11/.11;1}N (!l. Knowing: lutc Derelopment (y",tielf, 1 see and .1 find, New York, Basic Books.

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BL'I'ItK, D. (1985) The voices of women making meaning in mathematics', Journal ofEducation. 167.3, pp. 59-70.

DiwiosoN, N., AGM I N, L. and DAvis, C. (1986) 'Small group learning in junior highschool mathematics.. School Science and Mathematics. pp. 23-30.

GII I IGAN, C. (1982) In .4 Difliyent Ii)ice: Psychological Theory and Mmen's Development,Cambridge. MA, Harvard. University Press.

KiweNick. C.G. (1985) 'Women and men in the classroom: Inequality and its remedies'.On "kaching and burning, 1, pp. 18-25.

Lt wis. M. and SIMON, R. (1986) 'A discourse not intended for her: Learning and teachingwithin patriarchy'. Harvard Educational Review 56, 4. pp. 457-71.

MARcRoi. I, M. (1990) 'The pohtics of the classroom: Toward an oppositional pedagogy'.Neu. Directions.for 'Kuching and Learning, 44, Winter. pp. 61-71.

MArritoi i; M. (1971) On Caring, New York, Harper and Row.N( )i)i)INGS, N. (1984) Caring, Berkeley, CA, University of California Press.

i. A. (1979) On Lies, Secrets, and Silence: Selected Prose, 1966-1978, New York.Norton.

Roc it\ P. (1988) 'Student-sensitive teaching at the tertiary level: A case in

Proceedink of the livellth Annual Couference of the International Grourfor the Psychology ofMathematics Education, 2. pp. 536-43.

Roc to., P. (1990) 'Thoughts on power and pedagogy'. in Buit ION, L. (Ed) Gender andMathematics: An International Perspective, London. Cassell, pp. 38-46.

Roc:Fits. P. (1992) 'Transforming mathematics pedagogy, On 'Ruching and I..ea;ning. 4.pp. 78-98.

SimoN. R. (1987) 'Empowerment as a pedagogy of possibility'. Longuqe Arts, 64. 4.pp. 370-82.

Soi ow; D. (1982) Hole to Read and Do Proof An Introduction to Alathematics ThoughtProcesses, New York. J. Wiley.

Wii \oN. R.C. (1986. March/April) 'Improving faculty teaching: Effective use of studentevaluations and consultants . journal of Higher Education, 57,2, pp. 196-211.

185

Chapter 22

Gender Reform through SchoolMathematics

Sue Willis

Introduction

Feminists, from a variety of perspectives, have argued that aspects of schoolmathematics disadvantage girls. Some adjustnwnts to curriculum have occurred inorder to make school mathematics less overtly sexist and more attractive to girls,and some exciting and innovative curriculum projects have been developed.Nevertheless, certainly in Australia. and I believe elsewhere, these developmentstend to remain marginal to the mainstream curriculum, while major mathematics-curriculum refbrms occur with little serious attention to matters of gender. Whilemany feminists continue to critique school mathematics, on the whole themainstream mathematics curriculum remains unchallenged and relativelyunscathed by feminism. Even those of us who are most concerned with girls andmathematics have tended to focus on access as the problem, the task being 'to getit rather than change it' (Johnson, 1983, p. 15).

Changing Girls, Changing Choices

Reform programmes that target girls and mathematics often have focused onchanging girls by improving their achievementsittitudes, confidence, self-esteem,and risk-taking behaviour, or on altering their educational or career choices byimproving their participation in mathenutics or mathematically related occupa-tions (Kenway and Willis, 1993). The former include programmes intended toenhance self-e, xem with respect to mathematics, coupled with courses aimed atbuilding confidt ice mid competence in mathematics. Tlw latter include the useof role models, workshops designed to 'motivate mathematics, educational andcareer support and advice, all intended to encourage girls to make pro-mathematics choices. As important as many of these programmes have been andmay continue to be, in my view reforms that have as their main focus changinggirls or changing their choices will have lnnited success in enhancing female life

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Gender Reform through School Mathematics

options. Let me sketch out some of my reasons for this view.'First, they suggest a deficit view of girls that there is imething lacking or

in need of reconstruction even when the intention is to value and affirm girlsand girlhood. In order to argue disadvantage and justify intervention, we often findourselves emphasizing difference in a form that negatively compares girls to boys,women to men. In mathematics, girls lack self-esteem or confidence, they frarsuccess, they make poor choices, they do not understand or act in their own bestinterests, they achieve less well all relative to boys. Thus, we perpetuate certainways of viewing females and males which undermine girls and women. Indeed,girls have the right to question the extent to which many of us, who are supposedto have their best interests at heart, remain silent in the presence of such subtleundermining. Male skills and attributions are seen as the norm against whichfemales become the negative other to become equal is to become male.

Second, implicit in many of these approaches is the view that the relativeeducational disadvantage of girls causes their occupational disadvantage. That is,gender inequalities in the working world can be overcome if sufficiently many girlsleave school with the necessary mathematical skills and attributions to take theirfull share of the 'good jobs', or if they change their individual aspirations anddecisions. That girls should plan for economic independence and consider the fullrange of choices which may be available is clear. That their choices aboutmathematics should not be constrained by their gender is also. However, inattempting to encourage more girls and young women to participate inmathematics, we often paint a rather rosy pi:ture of the advantages which willaccrue to those who do so. For it is not at all clear that, if girls and young womenmade the same mathematics and career choices as boys and young men, theirearnings, status, and level of employment would be the same. Horizontaldesegregation of the labour market may be desirable, but as long as verticalsegregation remains in place, women are likely to be worse off th male-dominatedfields (Kenway Willis and junor, 1994).

My third concern is closely related. Encouraging in girl a sense of agency isessential, and it is here that the real strength of many programmes which endeavourto change the ways girls think about their lives is found. To suggest or imply thatgirls have a 'free (unconstrained) choice' to be and do anything, and that their'disadvantage' can be overcome by ch,mges in their individual attributions orchoices about mathematics is, I believe, dishonest and possibly even cruel. Theoptions of girls struggling with poverty racism, isolation or disability are oftenquite limited, regardless of individual qualities or decisions. Furthermore, all of ourchoices are constrained by cultural norms and values associated with gender, race,and class. Around the world. women are still regarded as the primary care-givers.and girls and women shoulder the major responsibility for tending the social fibric.In Australia, the struggles of women and men to adjust the ways that families arecared for have become a media event. Some progress has been made, but not asmuch as we had hoped, or as is often suggested. I believe that while weis feministeducators and mathematicians, acknowledge this, our :ducational programmesoften do not.

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We try to keep our messages to girls simple and often they become simplistic.This may explain why many girls don't find our arguments particularly persuasive.Recently, colleagues and I interviewed a considerable number of 14-year-old girlsand boys (Willis, 1991). With only a few exceptions, the girls saw their lives as tiedto fundamental conflicts of choice. They were planning their futures and expectedtheir lives as women to be segmented and manifold, while the boys were quitevague about their futures but perceived their lives more uni-dimensionally. Evenamongst the 'post-feminist' girls who insisted that 'everything was equal now' and'we can do whatever we like', there was almost always a 'but In essence, thesegirls were telling us that they knew the gender-imposed constraints surroundingtheir choices were real, not imaginary. Yet we are quite often exasperated whenwe try to convince girls that they can be engineers and they persist in missing thepoint and asking who is going to tuck in the baby When we tell them they cando anything, they suspect we mean that they are expected to do everything.

Fourth, approaches focusing upon 'changing choices' implicitly accept thehegemony of mathematics by which secure and high-status futures are seen as'naturally' accruing to the mathematically privileged. As almost any adolescent willtell you, the true power of school mathematics lies not in its utility in furthereducation and employment, but in its usefulness for accessing them. That is, amajor function of mathematics is as a de.Thao intelligence test and consequently, asa filter between school and sometimes quite unrelated further educational andoccupational opportunities. School mathematics, however, systematically dis-advancages students not only on the basis of gender, but also on the bases of socialclass, race, and ethnic identity (Walkerdine, 1989; Dowling, 1991). Thus, schoolmathematics is deeply implicated in reproducing social inequality. The problem forfeminists in mathematics education is that once having sold mathematics as aninvestment in the future, we can hardly attack the very exclusionary practices thatmaintain that investment potential.

Finally, interventions such as those I have been arguing against arc typicallymarginal to the mainstream mathematics curriculum and fail to challenge it in anysignificant way. Indeed, they may actually reinforce the notion that 'the problems'of girls and mathematics lie outsicle the classroom and school. Over the past threeyears, some colleagues and I have undertaken a series of case studies in Australianschools to investigate the reception of gender-reform messages by students andteachers (Kenway and Willis, 1993). Given the high visibility of the 'problem ofgirls and mathematics', and a push from government and industry to increase thenumbers of girls and women in mathematics, science and technology, we were notsurprised to find that female participation in these subjects was a major topic ofdiscussion in many schools. What was surprising, however, was just how little ofan effect this had on the practices of mathematics education. Only 3 very fewprimary or secondary school teachers believed that the mathematics curriculummight be gender-inflected and that reform was either necessary or possible, forafter all 'girls are as good as boys at mathematics and we don't make any distinctionsbetween them.'

Several years ago, television-advertising campaign designed to encourage

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girls o choose more and 'higher' mathematics was launched in several Australianstates. Put simply, the message to girls was that 'mathematics multiplies yourchoices', and to parents that you should not 'pigeon-hole your daughter'. Thiscampaign was read by many teachers we interviewed as confirmation that 'theproblem' lies with the poor choices that girls make as a result of social and peer-group pressure, home influence, or a lack of accurate information about theiroptions and futures. Thus, they appeared to hold the rather complacent belief thatthe curriculum was not an issue, and that it was equally appropriate 'for allstudents' in both content and pedagogy or, to the extent that it might not be, itwas inherent in the 'nature of the subject' and, therefore, unalterable. Somesuggested that the problem might be mathematics anxiety, low self-esteem, or alack of confidence, but had `no idea' how this might arise these characteristicswere, it seems. a corollary to being born female.

For these reasons, I believe that a focus on changing girls and/or their choiceswill not deliver justice in and through school mathematics. At best it will beinsufficient to the task, and at worst may prove counter-productive. I suggest thattbr feminist mathematics educators, a more productive focus can be found inchanging currkula.

Gender-Inclusive School Mathematics

Curriculum reforms designed to provide a different and better experience ofmathematics for girls may, themselves, be informed by different feminist (and non-feminist) perspectives. As a result, they will vary in their extent and form. In thissection. I will suggest some of the characteristics we might expect in a reformedmathematics curriulum.

'Thwards Non-sexist Curriculum Content

That school mathematics should not be sexist hardly needs to be said. There issonic acceptance that we should use non-sexist language and illustrations.Certainly in Australia, many teachers would routinely expect textbooks to pass the'non-sexist test'. That sonie of these changes are more apparent than real, and thatmuch of the .xxisin of school mathematics is more deeply embedded than can beaddressed by surfice changes in text materials is perhaps less well understood oraccepted.

Increasingly, we have conic to understand how mathematics curricula tend toemphasize the experiences, concerns, and interests associated with masculinityrather than femininity. Clearly, success with mathematical tasks will come easier tostudents who already have a grounding with the ideas underlying the task, or whoare familiar with the context in which the task is embedded. lieleen Verhage(1990) reported that she and her colleagues had been given the task of developinga mathematics curriculum which is both more realistic and more attractive to girls.By 'realistic mathematics education' is meant that the mathematics must be

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'derived from the reality around us [as well as be] applicable to this reality' andfurthermore, that students should 'rediscover (scraps of) mathematics andconstruct it themselves' (Verhage, 1990, p. 51). She observed a mathematics lessonon the study of symmetry and tessellation which drew on embroidery, a contextmore likely to be familiar to girls than to boys, and remarked,

[I]t was my impression that the girls clearly felt stronger during thi: lesson.They were at an advantage right away as it took the boys a while toactually get started due, for instance, to having trouble threading theneedle. The teacher told how, during the previous lesson, it had been justthe other way around, when it concerned building brick walls. (Verhage,1990, p. 66)

Examples such as this demonstrate the importance of context in enablingstudents to make the most of the learning opportunities provided in themathematics classroom. Since many of the contexts in which mathematics ispresented are more likely to centre in the lives of boys than girls, this gives boysa distinct advantage. Furthermore, such contexts must send quite clear messages togirls and young women about where mathematics came from, who it is for, andwhat its relevance is to them personally and to women collectively. Mathematicsproblems are likely to be either embedded in contexts more familiar andcomfortable to boys, er decontextualized and 'de-peopled' altogether. Neitherseems particularly friendly to girls.

Curriculum innovations such as that developed by Heleen Verhage seek toaddress this concern and are exciting and important. Unfortunately, the results ofthese efforts are often interpreted as simply adding to the repertoire of goodactivities and contexts from which particular mathematical concepts can bederived. Indeed, there is a widespread view that gender-inclusive schoolmathematics is largely a matter of making available the 'right' resource materials.As a teacher of mathematics said to me recently (private communication), 'I don'thave any problems with that [gender equity in school mathematics] if the resourcesare there'. The 'right' resources are balanced, presenting both sexes 'equally'. andproviding a range of examples interestin,, to all. According to many teachers,curriculum writers, and publishers, the 'right' resources also take subtle or indirectapproaches to gender equity. There are certain problems with this notion, towhich I will return shortly

Beiv Good at Alathematies

The work of Valerie Walkerdine and her colleagues has provided considerableinsight into the gendered (and classist) construction of mathematical ability. Theirearly work (Adams and Walkerdine, 1986) suggested that children learn thatSuccess in mathematics requires a passive approach involving rote-learning andrule-following, while what is seen as intelligent behaviour in mathematics is active,exploratory, and rule-challenging; the former behaviour often being associatedwith femininity, the latter with masculinity. Girls, even high achievers in

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Gender Rcform through School Mathematics

mathematics, are described as hard workers, plodders, capable, neat, conscientious,and helpful. Since these are not the characteristics which connote talent inmathematics, girls' early successes are not regarded as a sign of val. uit or even of reallearning, 'instead of thinking properly, girls simply work hard' (Adams andWalkerdine, 1986, p. 84).

Walden and Walkerdine (1986) suggest that the contradiction between thepractice of school mathematics which emphasizes rule-following, and theperception of the discipline that sees real mathematics learning as aboutchallenging rules, leaves girls in an ambiguous position. It sets up a conflictbetween what is regarded as necessary to achieve femininity and be a 'good girl'by doing what the teacher asks, and what is necessary to be regarded as 'reallygood' at mathematics. Vicki Webber (1991), in a case study of one of her adultstudents named Jenny, provides a powerful account of the manner in whichnotions of being a 'good girl' are embedded in the practices of the schoolmathematics curriculum. Jenny's hold on mathematics is very fragile and she feelsa fraud. As Webber (1991) remarks,

When the knowledge comes from outside yourself, you don't own it, youhave only a tenuous hold. You doubt the integrity of your own intellect... you are sure you will be found out ... you hold your breath and waitto see if you've got it right. Maths is an extreme example of waiting forsomeone else to say, 'you've got it right, good girl'. (Webber, 1991)

Jenny drew parallels between learning to be Catholic and learning mathematics:

Religious instruction, like maths, was filled with 'givens' and to questionthem was 'brazen, uppity'. Like the catechism she learned her maths 'byheart'. Like the Latin mass, maths was a secret language imbued withpower, and which could only be understood by certain chosen initiates.Maths too was ancient wisdom that one could not question, could notpresume to understand ... [As Jenny saysl these theorems had morcor less been handed down in the twelve tablets to Moses, and you've gotno right to think about them because they've been developed by all thesegreat minds. You've got nothing to contribute to this process. So you'vegot to learn them off by heart, to meet the requirements of being good,of being in the right.' (Ibid.)

For Jenny, 'maths was an altar with no steps up' and, in any case, girls could notbe altar-boys. Webber recounts,

Jenny felt in school that mathematics performance was closely associatedwith 'being a good thinker' and 'being mentally competent'. She had astrong sense in childhood that 'my mind wasn't good, but I can learnthings by heartind I can imitate quite well'. 'Real understanding' is thedomain of the great minds of men, women's understanding comes fromthe path of faith and devoted effort. (Ibid.)

This is not to suggest that girls are forced out of mathematics by some sort of

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patriarchal conspiracy. Teachers of both sexes are as caught up in these processesas are students who, themselves, are not passive in the process as they struggle tomake sense of conflicting messages. These contradictions, however, are part of awhole constellation of classroom practices which position males and femalesdifferently with respect to mathematics. As Walkerdine remarked, 'our educationsystem in its most liberal form treats girls "as if" they were boys. The "clever girl"is positioned as though she could and can possess the phallus, while she has tonegotiate other practices in which her femininity is what is validated.' (Walkerdine,1990, p. 46) The achievement of gender equity in mathematics education willrequire us to address the ambiguity involved in school mathematics for girls, aformidable task, and one we have hardly begun.

Tnvards a Different Conception (?fMatliematics

While the research is still fairly limited, considerable anecdotal evidence suggeststhat girls and young women need to experience mathematics as a way of knowingand finding out, as well as to understand its tentative, intuitive, exploratory,huimn, and therefore fallible, nature. Jenny, for example, tells us that it can bedifferent for her. After engaging in some problem-solving tasks, she said,

After that I understood that this is how maths has developed throughhistory. Mathematics is a shorthand for describing a real, meaningfulprocess ... I knew what that formula was about. It was an expression ofwhat 1 had actually done ... I know now where a formula conies from,that no matter how complicated it comes from real problems in the realworld. (Webber, 1991)

It seems that our students need to know that mathematics makes sense, that theycan make sense of it, and that they can work it out.

Several years ago, the Australian Government set up a programme whichfunded the development of 'gender-inclusive' teaching materials in mathematicsand science. Mary Barnes was commissioned to produce a set of student textmaterials for teaching introductory calculus at the secondary. level. As with theNetherlands project on realistic mathematics education, the goal was to model acurriculum that was non-sexist, used a more inclusive range of contexts, andacknowledged and incorporated women's contributions to both mathematics andsociety. It was to be rigorous and intellectually demanding, acknowledging andvaluing some of the power of the 'disemheddedness' of mathematics, while at thesame time embedding it in social concerns and people-oriented contexts.Furthermore, it was also to present mathematics as making 'human sense', non-arbitrary and fallible, produced by and for people, and having aesthetic, cultural,and instrumental importance. A rather ambitious agenda!

According to Barnes, the course stresses mathematical modelling and thesolution of realistic problems, with less emphasis on traditional applications fromthe physical sciences and more emphasis on issues of importance in students' lives.It invols es students working collaboratively to investigate problems, explaining and

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clarifying their thoughts in the process, while at the same time sharing insights,experience, knowledge, concerns, and confusions. The goal is for students tounderstand that mathematics should make sense and that they, themselves, arecapable of deciding when something does or does not make sense.

The materials are now published (Barnes, 1991, 1993). and the indicationsregarding their use are encouraging. Although some students in the trials did notrespond positively to the changes, most did. Overwhelmingly, their commentsreflected an increased confidence in the quality of their knowledge. as well as afeeling of control over their own learning. They commented about under-standing:

Girl My dad said he never really understood what it was about ... I

understand what it is about and I can do the problems whereas he can dothe problems without really knowing.Boy When fou get in groups you learn better, because you are mixingup your knowledge.Boy The course is much easier to understand because we know why.Girl They did make sense unlike other things we were doing in maths ...we were able to understand where they came from ... I enjoyed it.

about taking responsibility for their own learning:

Girl At first the groups did what she [Mary, the author] said, you knowhad one of us writing and one timing ... But then only one person hadthe ideas written down, so if you looked at it at home you might notremember. So now we talk about it and decide what we all want to sayand then we all write it down ... Sometimes we argue but not usually.Girl You have to make sure you stop and say in your group. Well, what'sgoing to happen? and make sure everyone understands the why and thendo it ... We just plugged it in [the computer] for a while and we didn'tlearn anything. But we realized.

and about purpose:

Girl The best thing about them is how they related to real life. Most ofthe problems were practical. It makes it easier.Boy It's more real life so you had to think about it more. There were noreal formulas.

One teacher, commenting about his female and male students, said, 'I havebeen so overwhelmed ... by the insight they have about graphs and the derivativefunction and what the derivative does ... They are just so much more powerfulthan the previous group. Their ability to justify things like the chain rule wasbreathtaking.' To be able to describe students as 'powerful' in doing mathematicsis surely what we would like to achieve. Quite a few teachers commented ongender difkrences in response to the approach. Girls, they variously claimed.

were more ready to share and to talk freely;

Sue Willis

were enjoying this work although they hadn't in the past;work better in groups;had more self-discipline;were more worried, but would come and see you about problems;were more creative;wanted to know why; andfound the examples more interesting [than girls studying calculus have inthe past].

Boys, by contrast, teachers said,

were less ready to share their ideas;tended to waste more time;tended to be domineering, but now realize that other people have positivethings to say;were more easily distracted; andwere happy to just do it [rather than know why] and come up with theanswer and think that's great.

It seems that in these classrooms, teachers and students of both sexes arebeginning to change their perceptions about what mathematics is, who producesit and how, and what it means to learn mathematics and be good at it. Making suchcurricula a reality in schools is an ambitious goal, but were we to be successful,surely it would be both more rewarding and just. The fundamental question,however, is would it be emancipatory? Probably it would, but for it to fully reachits emancipatory potential. I believe it must be accompanied by explicit attentionto gender and power relationships, as well as to how mathematics is implicated inperpetuating inequalities in these relationships.

Socially Critical Curriculum in Mathematics

Earlier I suggested that, to the extent that more 'gender-inclusive' curricula aredeveloped, there is a preference for 'subtle' and 'indirect' approaches where thegender 'agenda' is not made explicit. As director of the programme through whichMary Barnes' materials were generated, I was constantly warned of the dangerthat, by being either too direct in our approaches, or too explicitly feminist, ourefforts might be rejected by teachers. This dislike of direct approaches seemed tolie in their potential to be disruptive to the good progress of the class 'as a whole'.Potentially disruptive approaches included those which confront sexist assump-tions and behaviours explicitlyind thus might offend some teachers, students orparents, or, perhaps, disturb or delay the achievement of the primary goal, that is,teaching 'the curriculum'. The argument was that nothing would be achieved ifthe materials were not used because teachers or students found them toothreatening. Since the materials produced through the programme were intendedfor mainstream use, we succumbed and so few of our products were explicitly

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feminist. However, as illustrated by the following three examples, certain dangersare inherent in such indirect approaches.

Paul Dowling (1991) describes a well-known 'counter-sexist' mathematicstextbook. Female names and pronouns are re ularly used in conjunction withtraditional male occupations and roles, and vice versa, but this is done withoutcomment and the resulting dissonance is 'occasional and never explored'(Dowling, 1991, p. 3). Dowling suggests that this often becomes a target forhumour, which may be counter-productive. He also recounts a story relating toan activity in the textbook which involves 'famous people', two being women,George Eliot and Florence Nightingale. The illustration was of a statue of GeorgeEliot, with her name clearly printed below. One (female) student he spoke toassumed that the statue was of Florence Nightingale and had been misnamed. Thisstudent dealt quite readily with the dissonance involved in having a female figuregiven a 'male' name by explaining it to herself as an error. For this student, at least,the textbook was not at all disruptive.

Heleen Verhage (1990) recounts how a student in a lesson on symmetry andtessellation commented, 'This isn't math. This is sewing class', to which theteacher replied, 'When you have to draw a circle you don't call it drawing class,do you?' (Verhage, 1990, p. 66) The teacher's answer is valid, as is the underlyingpoint that school mathematics is a social construction. Such occasional andincidental rebuttals, however, are unlikely to challenge the student's perception ofwhat mathematics is, what counts as mathematics, and, therefore, what it meansto be good at mathematics.

Students and teachers were interviewed during the trial period of MaryBarnes' calculus course. One common early reaction on the part of teachers wasthat the materials required more reading than students would do and that, inparticular, this would disadvantage their weaker pupils. When asked to gathermore inforn ,tion about who exactly was and wasn't doing the reading, and whowas complaining, the teachers returned rather abashed. They had failed to noticethat the original complaints had derived from a vocal minority of high-achievingboys, and a few high-achieving girls, who felt that there was more explanationprovided than was necessary and it slowed them down. The students whom theteachers regarded as 'less able' insisted that they did the reading, and as one studentfiercely demanded, 'Don't you remove a single word.' Moreover, it waspredominantly the high-achieving male students who were most reluctant toparticipate in the group work that was an essential component of the course. Theseobservations demonstrate how, unless we are vigilant, those already privileged inmathematics can continue to control and define what constitutes mathematics inschool.

The above examples demonstrate that gender-inclusive materials do not bythemselves ensure a gender-Mclusive curriculum. In each case, interpretationswere based on existing frameworks and, in the absence of any disruption, thepotential for significant change was not there. I would argue that the effect of theintervention would be more powerful in each instance if the agenda were madeclearer. What is needed is a more socially critical mathematics curriculum in which

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mathematics is regarded both as a powerful tool for critique, and also as an objectof critique. Allow me to provide just a glimpse of what this might mean.

Women have traditionally sold their skills cheaply because they and otherseither do not recognize them, or may even denigrate them (see, for example.Pocock, 1988; Holland, 1991). Often, women have not been prepared to believethat they had any skills worth possessing unless they were male-defined. This is astrue of their mathematical labour as it is of their labour in general. As Mary Harris(1989; see also this volume, Chapter 9) has pointed out, designing the lagging fora pipe that turns at right angles is an engineering feat requiring significantmathematics, while designing a sock for the heel of a foot is a trivial knitting taskrequiring insignificant mathematics. However, as I have already suggested, simplyensuring that socks and pipes, embroidery and bricks are equally represented in amathematics curriculum is unlikely to be enough. The question of why sewingisn't mathematics, while bricks and circle drawing are, needs to be addressed withstudents. Indeed, Heleen Verhage (1990) raised the vexing question of whetheractivities such as embroidery are emancipatory or merely serve to confirmtraditional role patterns. In my view, the reception and effect of such activities islikely to be influenced by the extent to which their gendered nature is madeexplicit. As a colleague and I have argued elsewhere (Kenway and Willis, 1993),'home still signifies the place of the female, apart from the public and importantworld of the male. By leaving 'home' out, we may actually emphasil:e itsunimportance and hence female unimportance, thus reinforcing rather thanreducing gender stereotypes. As I suggested earlier, to be equal is to be male.

A recent Australian newspaper advertisement promoting a city building projectwhich would generate 'thousands ofjobs' provided a series of about fifty male andfemale figures labelled with the name of a job and the number of positions to begenerated if the project went ahead. In the advertisement, 8.3 per cent of the totalnumber ofjobs were represented by female figures, and not a single one was of non-European origin. Women and minority Australians remain the silenced unem-ployed. A close look at the poses and style ofdress ofthe figures also revealed appallinggender, class and racial stereotypes. Images, such as these, do the most damage whentheir sexism, classism, and racism go unnoticed at the conscious level and are thusabsorbed as natural or accurate descriptions of the world. The mathematicscurriculum can make students aware ofsuch things in order to deal with them. Issuesrelating to unemployment and 'women's work', both paid and unpaid, mightbecome the focus of any one of a number of short or extended projects inmathematics. We might, for example, ask students to use their mathematics toinvestigate the consequences on women, both individually and collectively, as well ason the' entire community, of not defining domestic labour as 'work'.

Elsewhere (Willis, 1994), I have described an Australian series of mathematicstextbooks which on the surface appeared to have nude considerable ,.'ffOrt to b,'non-sexist by using gender-neutral language and illustrations, and by hav ing girlsand boys equally represented and equally and actively involved in 'real-world'mathematics. However, an analysis of the representations of adults in the textsprovided quite a different picture.

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Almost all women (except teachers) re referred to as mum or grandmaor the butcher's wife, almost always cooking or shopping, and rarely givena name. Men, on the other handire occasionally called dad ... but aremostly given names and described in terms of their job. Men have andhandle more money. To what extent do these books help to convinceboys and girls that their mothers should be at home and available to helpout at school, and their fathers should be supporting the familyfinancially? And do they imply that girls and boys should chose to emulatethese more 'suitable' mothers and fathers? While mathematics is nowregarded as important for all students, perhaps lower levels of mathematicsare sufficient for women's work. (Willis, 1994)

One approach to dealing with such materials nnght be not to use them. Is itconceivable, instead, that we might use them as an object of study, that studentsmight be asked to investigate how men and women are constructed in theirmathematics textbooks? Might not they lso use their data collection and analysisskills to investigate who controls the classroom agenda and whose interests are metby the materials they use?

I commented earlier on an Australian television campaign aimed at encourag-ing more girls to participate in mathematics durthg their upper-secondary years.The advertisements asserted that you have 400 per cent more choices if you domathematics. Unsuccessfully, I tried very hard to find justification for such figuresand, indeed, the explanation of the data which was provided to me showed .1 clearmisuse of mathematics. Were we seriously claiming that 80 per cent of all jobs, ofevery kind ofjob for every kind of girl, needed mathematics in the final two yearsof schooling? Of course not. By increasing participation in mathematics, will webe increasing the number ofjobs available? Again, the answer is no. Most popuiaranalyses I have read which attempt to forecast where the jobs will be in the futuresuggest that mathematics will not be central in doing themdthough it may wellbe crucial for getting them. Neverthe.ess, the advertisements were 'successful'according to the advertising company's criteria, that is, there was good 'saturation',

lot of people remembered having seen them. But what, I wonder, was the eflixtof the campaign on the morale of those students whom school mathematics hasnot served well? And what was the effect on young women and men of theportrayal in the advertisement of girls in a foetal curl squashed into boxes lookingsad and oppressed :don't pigeonhole your daughter')? Had these images of youngwomen appeared in a men's magazine, we would have been scandalized.

To those feminists who were critical of these campaigns, the only winners inthis campaign were the employers, for they would Dow have more choices. Severalcolleagues tried to have the 'voice over changed before the advertisement wasbroadcast to remove the reference to '400 per cent' on the grounds that it wasdistorting and misleading. They were told that the advertisement would Lick'punch' without the numbers, since 'people find figures very persuasive'. Thus, thecampaign itself provided an example of the use of mathematics to distort,intimidate, and mystitV. Mathematics was used on girls, not .14 them. This is

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precisely the sort of media claim that we would hope a quality mathematicseducation could enable students to critique. A productive use of this advertisementwould be to encourage students to deconstruct it. Students should be able tointerrogate the claims, asking what the 400 per cent actually refers to and how itwas determined, thus evaluating the sexist and classist assumptions upon which theadvertisement was based. This task would certainly not be beyond the capabilitiesof most 15-year-olds and has enormous emancipatory potential. Of course, theinevitable questions which would follow are: What is it about mathematics thatencourages people to suspend their common sense and hence allow such materialto go unchallenged and whose interests are served when such programmes 'go toair'? The difficulty in adopting such approaches is in, finding the balance betweenbeing too disruptive and not being disruptive enough. In the former case, studentsmay feel so uncomfortable that they resist completely and are unable to engage.In the latter case, they may comfortably accommodate new ideas within theirexisting frameworks and thus find no need to change their thinking.

Conclusion

The task of mathematics education should, explicitly and unapologetically, be theprovision of a rewarding, justind emancipatory educational experience for girls.And the justification for doing so is precisely that it is rewarding, just, andemancipatory. However, this should not come about by providing alternativecurricula for girls. Everyone's attitudes and beliefs about what constitutesmathematics, what counts as valued knowledge in mathematics, and what it meansto be good at mathematics must change. Consequently, students of both sexes willneed to experience mathematics differently than they do currently.

Mathematics curricula should aim to enhance female life options by makingchoices more fully and equally available. Girls and young women need to understandhow their attributions and choices about mathematics, as about all else, are sociallyconstructed and constrained. A gender-expansive and more critical mathematicscurriculum would address the ways in which mathematics is used to intimidate andmystify, thus confronting the issue of gender formation in school explicitly. This, inturn, would enable both girls and boys to understand how they arc positioneddifkrently with respect to mathematics, to identify the processes by which genderedpatterns of ochievement and participation in mathematics are produced andnaturalized, and to recognize the effect of these processes on their futures. Over adecade ago, Jean Blackburn, a former Australian Educator of the Year who first usedthe term 'sexually inclusive curriculum' in the early 1980s. suggested that one role ofeducation is to help young people 'to understand and reflect on what is, on how itcame to be that way and on what they want to do about it, both at the level of theirown lives and in relation to social action' (Blackburn, 1982, p. 10). The purpose ofstudying gender relations within a socially critical framework is not to point andmove girls (or boys) in certain directions, but rather to widen the range of positionsavailable to them both now and in the future.

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Note

These points Were developed in collaboration with Jane Kenway. Deakin University,Australia and the Girls Unit of the Education Department of South Australia.

References

ADAMS, C. and Wm KFRDINI., V. (1986) Investigating Gender in the Primary School, London.Inner London Education Authority.

BARNES, M. (1991, 1993) Investigating Change: An Introduction to Cakulus _for Australian

Schools, Units 1-10 and Teachers' Hanlbooks, Carlton, Victoria. CurriculumCot poration.

BI MI:BURN, J. (1982) 'Some dilemmas in non-sexist education' , The Secondary 'Mather, I.

pp. 10-1 I.Dowi INC, P. (1991) 'Gender, classmd subjectivity in mathematics: A critique of Humpty

Dumpty'. For the Learning if Nlathematics, 11, 1, pp. 2-8.HARRIS, M. (1989) 'Textiles and maths'. GEMS, 1,4. pp. 37-9.Hon ANI ), J. (1991) 'The gendering of work:', in HARRIS, M. (Ed) Schools, Mathematics and

librk, London, Fahner Press, pp. 230-52.JOHNSON, R. (1983) 'Educational politics: The old and the new', in Woi ir, A. and

DoNrki o, J. (Eds) Is There Anyone Here from Education?, London, Pluto Press,pp. 11-26.

KI.NWAY, J. and Wti is. S. (1993) Ming 'Thles: Girls and Schools Changing their Ways,Canberra, Department of Employment, Education and Training.

KUNWAY, J., WII i IS, S. and JUN014., A. (1994) Critical Visions: Curriculum and Policy Rewriting

the Future, Canberra, Department of Employment. Education and Training.Poco( :K, B. (1988) Demanding Skill: 1Viunen and li.chnical Education in Australia, Sydney,

Allen and Unwin.VI.RI H. (1990) 'Curriculum development and gender', in BUItTON, L. (Ed) Gender

and Mathematics: An International Perspective, London, Cassell, pp. 60-71.WAI DUN, R. and WAI KERIBNE, V (1986) 'Characteristic:;, views and relationships in the

classroom', in BUR I ON, L. (Ed) Girls into Maths Can Go, London, Holt, Rinehart andWinston, pp. 122-46.

Wm KI RDINt., V. (1989) Counting Girls Out, London, Virago Press.WALKER !ANL, V. (1990) Schoolgirl Fictions, London, Verso.

it, V. (1991) 'Dismantling the Altar of Mathematics', Unpublished paper.Wit its, S. (1994) 'Mathematics: From constructing privilege to deconstructing myths', in

GASKI I I , J. and WII I !MK Y, J. (Eds) Gender In /j.orms Curriculum: From Enrichment to

liansformation, New York, Teachers College Press.

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Chapter 23

We Don't Even Want to Play:Classroom Strategies and Curriculumwhich Benefit Girls

Joanna Higgins

Introduction

Mathematics games, puzzles, and constructions are commonly used along withother independent activities in the classroom to familiarize students with particularmathematical concepts. The number of players, the amount of time spentparticipating, and the range of skills and concepts explored will vary in any oneactivity. In this chapter. I will discuss the ways in which boys and girls interactduring these independent activities. I will highlight the fiictors which contributeto a differential experience between girls and boys, in which the former arcdisadvantaged as mathematics learners. I will suggest that the manner in whichmai,:s respond to the teacher's chosen curriculum shapes the learning experience.My analysis of interactions between children playing mathematics games will showhow girls are disadvantaged in this context. Finally, I will propose classroomstrategies and curriculum which are more likely to benefit girls.

Gendered Achievement and Order in the Classroom

Clark (1990, p. 2) has pointed out that 'the primary [school] years are crucial inthe construction of gendered achievement and aspirations and that the neglect ofthese years in most gender equity programmes is a serious problem which must beredressed'. When trying to identify micro-processes operating within the class-room that offer potential changes to benefit girls, it is important to make thegender dynamics of the classroom visible for teachers and educators. Alton-Leeand Densem (1992), for example, have drawn attention to phenomena that areoften hidden from both teachers and classroom observers. They suggest that theexistence of subtle gender and racial bias in the classroom requires the debate aboutgender-inclusive curriculum to move beyond issues of access and participation.More specifically, it needs to contend with the hidden processes of classroom

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structures, as well as examine the differences between the intended, theimplemented, and the perceived curricula.

Power structures of the classroom influence student behaviour as well as shapethe learning environment in ways which benefit boys more than girls. Theteacher's predominant concern is with maintaining order to facilitate complex,autonomous, learning activities. Fennema and Peterson (1985) identify inter-actions with the teacher as key factors in the development of autonomous learningbehaviours necessary for high-level cognitive tasks such as those in mathematics.They note that the teacher is less likely to direct girls back to a task because theirbehaviour is often seen as less disruptive than that of boys. Furthermore, malesreceive more disciplinary contacts, and more praise, when on-task. This suggeststhat boys' le.arning is facilitated by the teacher's concern with maintenance oforder. Such structures in the classroom give boys an advantage in the learningsituation. An important f-actor which requires consideration is the manner inwhich girls perceive the teacher's actions. The fact that the teacher's attention isdirected away from the girls' quiet learning to boys' off-task behaviour simplyreinforces the male advantage. Girls, as a result, arrive at the belief that theirlearning is less important.

Freedom of Choice

The provision of choice has often been championed by teachers as a means ofmeeting the individual needs of learners. In practice, choice refers to the provisionof a range of options which relate to a set of learning goals and from whichchildren are expected to select. However, the way the teacher structures thelearning environment constrains this freedom of choice. Similarly, learningoutcomes are influenced not only by teacher assumptions about intended learning,but also by gender dynamics which are often not taken into account. Manycommentators have argued for a reconsideration of the child-centred model onwhich such classroom practices are based. Clark (1990, p. 26) points out that'child-centred practices can allow teachers to avoid facing this responsibility [forproviding equitable learning opportunities] by allowing teachers to believe thatchildren can "freely" choose and that we can be neutral'. The provision of freedomof choice needs to be re-examined, not only for the reasons I have already stated,but also because boys choices are commonly treated as being of more value. Thisresults in boys 'owning' certain types of activity.

Having Access

Boys' ownership of certain activities is illustrated in the following transcript of aclassroom segnwnt in which girls are simply given access to the blocks.' Genderdynamics are evident during the construction of a three-dimensional model of abuilding. The boys' possession of this activity proves to be too much for the girls

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in the end, and the learning opportunity intended by the teacher is sabotaged.Rather than persevering in their own model building goals, the girls learn tosuccumb to the boys' demands.

Two groups are working at the same activity the blocks. One is a groupof boys, the other a group of girls. The boys need more blocks to maketheir model bigger. While searching around for these, they look across tothe girls' model.

Michael: We need a block like you've got.He jumps across to the model and takes one. The girls try unsuccessfullyto stop him and this leaves a gap in their model.

Shaun: We might build a huge place and we might use up all theblocks.

Martha: No, Shaun.Shaun threatens to grab some more of the girls' blocks.

Martha: No Shaun, we might bash.An observer asks Philip to tell her about the model that the boys havemade.

Philip: It's a big castle just for people and people can go up there,[pointing], and down there, over there on the top to look outfor some gold.

She comments that she heard him asking the girls if they would like tojoin.

Philip: We were going to join them, but they didn't let us.He points to another part of the model.

Philip: And cars who have broken down would get fixed here.She asks him why they wanted to join up with the girls.

Philip: We wanted to join up so it could be really really big.Terry comes over so she asks him to tell her about the model.

Terry: I made a big building so you could jump across the sea.The observer moves across to the girls' model and talks with Martha.

Martha: It's a school. These are the seats [pointing]. That's theplayhouse. That says 'school today'.

Philip comes over to the girls' model.Martha: Would you leave it, Philip.

The girls' model is fast disappearing. The observer asks all the girls whythey didn't want to join up with the boys.

Martha: Just didn't want to.Nadine: They always say if you try and bust up ours we'll bust up

yours.Michael comes over to get some more blocks. The girls finally give intohis demands.

Nadine: You can have these blocks if you want 'cos we don't evenwant to play with them.

It is evident that gender dynamics intervene not only in the structure of this

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learning opportunity, but also in the intended learning outcomes. The boysactively engage in model building and achieve their goal of making 'the biggest'.The girls, on the other hand, learn only to give up their own goal in the face ofboys' aggressive demands. This example suggests that the notion of freedom ofchoice as a means of achieving equity in learning is an illusion. Gender dynamicswill commonly intervene in such experiences, and teachers must adjust for this bydevising techniques that facilitate girls' learning.

Being the Teacher

Walden and Walkerdine (1986) have argued that issues of dominance andsubordination can be seen in the social relations of the classroom. For example,Bird (1992) found that girls seem to be much more willing than boys to assumethe role of teacher. In my own observations, there was an obvious diffc rence in thenature of the leadership roles taken up by girls and boys.

Four children are participating in a board game as part of theirindependent mathematics activities. It is designed to help children learnhow to order in numerical sequence, as well as understand the 'one morethan' relationship.

Anne: Whose turn?Mary: Martha's.Tim: Mine.Anne: Tim.Martha: You had lots of turns.

Tim takes his turn.Anne: Go Martha. One more to go to win.

Martha has her turn.Tim: I've got two more to go.Anne: Your turn.

Mary plays.Martha: My turn.Mary: Yes. Four.Anne: Martha, don't cheat. It was Tim. Martha your turn. Right

now Tim.Martha: Right your turn.Tim: Supposed to be my turn.

Tim knocks his card off the board.Anne: Don't Tim. Hey cheat. [to Mary]Martha: Your turn, Timmy.Tim: My turn.

Anne mixes up the cards.Anne: Now you can't get two fish to go there. Tim, your turn,

Tim has his turn.

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In this game Anne took on the role of the teacher. She took control of thegame by directing the turns and by encouraging players not to abandon hope whenthey seemed to be losing. She also monitored the game for cheating, and whenthe only boy in the game misbehaved, she reprimanded him. Clark (1990) hassuggested that boys and girls have available to them different ways of beingpowerful. Boys have access to direct forms of power such as sexual harassment ordirect challenges to the authority of the teacher. Girls are more likely to gain powerthrough indirect means, for instance, by b,ing a quiet cooperative model pupil, orby taking on the role of teacher.

Commentators such as Clarricoates (1987) have suggested that girls avoid risksin their learning by adopting feminine roles. hi this context, assuming the role ofthe teacher should be interpreted as the child conforming to the relativelysubmissive expectation of 'helpful pupil'. Encouraging girls to behave in thisfashion merely sustains them in the traditional feminine position of subordination.However, arguing that 'things are not always what they seem', Bird (1992) hassuggested that the girl who plays 'teacher' is instead taking up a powerful positionby claiming the authority of the teacher. Similarly. Walden and Walkerdine (1986,p. 125) suggest that 'by being positioned like the teacher and sharing her authority,girls are enabled to be both feminine and clever; it gives them considerable kudosand helps their attainment'. As teachers, we need to critically examine the socialcontent of the roles we provide for girls in a mathematics learning activity. Weneed to ensure that girls are adopting roles which will enhance their learning ofmathematics The creation of all-girls groups will not necessarily achieve this onits own.

Girls and Competition

Children's own gendered expectations about game-playing behaviours have asubstantial impact on participation patterns. In particular, it is difficult for somegirls to respond to competitive situations in the classroom. The two transcriptswhich follow demonstrate the contrast in gender-related reactions to a competitiveenvironment. The activity is an adaptation of the board and card game 'Memory'which has been designed to help children develop a concept of numerical order.5-year-old children, in groups of four, attempt to be the first to cover all themarked spaces on their board with a card. A space can be covered if a child turnsover a card from those spread out fice down which shows a set containing onemore element than the set shown on the corresponding space of their board.While they engage in this undertaking, the teacher is involved in an instructionalactivity with another group.

Philip, Terry, Shaun and Michael are sitting playing a board and cardgame. Philip takes his turn.

Philip: You've got four points. I've got four points too. Both got fourpoints.

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Terry has his turn, but conceals his card instead of showing it to the othersas is normally done in 'Memory'. One aspect of the game is rememberingwhere particular cards are positioned among all the cards when they areplaced face down. None of the other players object to his cheating.

Philip: Oh, you got five points too.Terry: Have you seen any of mine?

Terry tells an obmver that you have to sneak a look at the cards.Terry: Do you have six points?

Philip counts.Philip: Yeah, I got six points.Terry: Who's won?Philip: I got six. Let's play again. Now people, help me turn them

over.The boys then start the next round of their game.

The above scenario documents the =liner in which the competitive aspectof this game with its goal-orientation is compatible with the boys' learning style.They appear to be motivated by the collecting of points and the chance to win thegame, as shown by their enthusiastic and immediate playing of the next round. Forthese reasons, teachers often use competitive activities to minimize the potentialfor boys' disruptive behaviour.

We see a marked contrast to this in the responses of a group of girls playingthe same game.

Anne: Martha, your turn.Martha: You had one in your hand.Anne: Don't let Mary see, Martha.Martha: I won, I won, I won.Anne: You lose, Mary. Three more turns. I won after Martha.Martha: I won first.

At this point Mary starts crying.Mary: You're cheating.

Martha gets up at once to comfort Mary by putting her arm around her.Martha: I didn't say anything.Tam: It's only a game.

Mary is still crying loudly. At this point the teacher comes over.Mrs S: Oh Mary.

Mary tells her that the others cheated.Mrs S: I think you and Anne should play somewhere else as you were

nearly in tears the other day as well.

The girls are concerned both with monitoring the procedures of the game, andwith supporting all members of the group. When one participant becomes upset,the others comfort her and are diverted from continuing with the game.

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Participation Patterns

When we compare the respective responses of each gender group to this game, wesee that there are differences in attitudes, interactions, and participation patterns,and that these all influence learning outcomes. The words chosen by each groupserve as a window to the attitudes involved in the playing of the mathematics game.The boys appear to have a much more competitive attitude, indicating theirconcern with winning as the primary goal of their interaction. Most of theirconversation revolves around how many 'points' each has accumulated. The girls,by contrast, talk of 'turns'. This indicates their preoccupation with the proceduresand process of the game, rather than with the goal of winning. Thus, while theboys learn to win, the girls learn to play the game. It is interesting to note that inall of the activities described here, learning the mathematical concepts involved isalmost incidental.

An additional aspect of gender differences in game-playing can be witnessedin the way each group responds to cheating. In the female group, Anne in her'teacher role' instructs Martha not to let Mary see the cards. This is followed byMary's accusation that Martha is cheating when Martha claims she has won. Incontrast, Terry's query about others having seen his cards, as well as his assertionthat it is necessary to sneak a look at the cards of other players. indicate that theboys harbour the belief d-tat cheating in order to win is a necessary part of playinggames such as 'Memory'.

The overall participati )n patterns of the two groups can also be seen to differ.The interactions among tht boys appear to be less structured in terms of rules andturn-taking than that among tne girls. Consequently, the boys are able to achievethe goal of rapidly placing cards in order to show the one-more-than relationship.The girls, however, seem to lose sight of this goal altogether. In their group, thestructuring of the game procedures is provided through someone adopting the'teacher role', while in the boys' group this does not occur. The boys appear toengage in self-monitoring of their progress as they work towards the goal ofearning six points. On the other hand, the girls seem to be more concerned withmonitoring each other's behaviour and conformity to game procedures. As aresult, learning outcomes for the riirls are primarily social rather than mathematicalin nature. In sum, since gender differences in learning styles influence learningoutcomes, girls may fail to engage with the intended mathematical concepts incompetitive game situations.

Conclusions

It is imperative for teachers to realize that gender expectations and genderdynamics may intervene in the intended learning experience. In this way, girls'learning can be severely disadvantaged. As has been illustrated in the precedingdiscussion, the importance conveyed upon the maintenance of order in theclassroom has several unintended results. One is that boys' learning styles and

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dominance are reinforced through the provision of competitive activities designedto minimize their disruptive behaviour. Another is that because they are not asdisruptive in the classroom, females receive less teacher attention and constructiveconcern than their male counterparts. On a related note, females are furnishedwith less encouragement to stay with the learning task. Taken together, the impacton female learning is very serious.

All of this suggests that if girls are to benefit from independent mathematicstasks, more cooperative, as opposed to competitive, activities need to be offered.While only a first step, such an approach would better match girls' observedlearning styles. A more carefully selected range of choices should function to helpgirls find activities in which they can achieve success. Furthermore, the provisionof a sufficient quantity of equipment would reduce the possibility of girls being'squeezed out' by boys. In this way boys 'ownership' of particular activities couldbe diminished.

It may be necessary for teachers to reconsider the assumptions on which theprovision of teacher-independent exercises are based. As the preceding examplesshow, gender dynamics will often intervene in activities which are grounded in thechild-centred principle of choice. Neither access, nor opportunity, will be equalunless teachers restructure the learning situation to support the needs of both girlsand boys. Girls need to be encouraged not only to persevere in problem-solving,but also to engage in risk-taking behaviour.

In order to support all learners, teachers need to monitor the groupinteractions which take place during independent mathematics activities and tomanage gender dynamics. An important means of such support is the teaching ofsocial skills that are crucial tbr learning and necessary for engaging in cooperativeactivity The skills for working in groups. for listening to others, for generatingideas collaboratively, and tbr participating in reflective discussion cannot beassumed inherent in young children. Furthermore, these skills are critical in thefostering of mathematical development. Designing a learning environment inwhich all students engage fully with particular mathematical concepts requires atleast as nmch attention to gender dynamics as to mathematics itself This isparticularly true when independent activities are used for pedagogical purposes.

Note

1 Obwrvat.ons cited in this chapter are drawn from research for tny masters thesis(Higgins. 1991). Data was gathered over a thirty-week period during 1989 usingstandard ethnographic practices.

References

AI \ -LI i . A. and Di Nsi M. I (P)92) 'Towards a gender-inclusive school curriculum:Changing educational practice'. in Mu )1)1 u.o .... mmmd ji A. (Eds) Mmeu nd

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_Panful iligzitts

Education in .4otearoa 2, Wellington, Bridget Williams Books Ltd.. pp. 197-220.BIRD, L. (1992) 'Girls taking posifions of authority at primary school', in Munn i roN. S.

and j()Ni..S, A. (Eds) Women and Education in ,4otearoa 2, Wellington, Bridget Williams

Books Led.. pp. 149-68.CI ARRI(.0.A.Iis, K. (1987) 'Child culture at school A clash between gendered worlds?'

in Poi .tuw. A. (Ed) Children and Their Primary SMools, London, Falmer Press,

pp. 188-206.Ca ARK, M. (1990) The Great Gender Divide. Gender in the Primary School. Melbourne.

Curriculum Corporation.DAVI B. (1988) Gendo. Equity and Early Childhood, Commonwealth of Australia.

Curriculum Development Centre.NNI NIA. E. and 131,11.100N, P (1985) 'Autonomous learning be.haviour: A possible

explanation of sex-related differences in mathematics', Educational Studies in Mathe-

milks, 16,3, pp. 303-20.Hic,t;INN, J. (1991) 'Mungapungas and Brooming Revealed: Independent Learning in Early

Mathematics', Unpublished masters thesis. Victoria. University ofWellington.

WAI DrN, R. and WAI KI iwixi, V. (1986) 'Characteristics, views and relationships in the

classroom', in BUR I Y.. L. (Ed) Girls Into Maths Gui Go, London, Holt, Rinehart and

Winston, pp. 122-46.

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Chapter 24

Moving Towards a FeministEpistemology of Mathematics'

Leone Burton

Introduction

Received science has been criticized on three grounds from a gender perspective.The first is its reductionism and its claim to be objective and value-free (forexample, Harding, 1986, 1991; Keller, 1985; Rose and Rose, 1980). Second, theconventional style of learning and teaching in science, its pedagogy, has beenchallenged. It is suggested that enquiry methods used by scientists are oftenintrusive and mechanistic, separating observer and observed, and reinforcingcompetition. Further, these methods are presented not only as 'correct' but alsoas the only way possible (for example, Kelly, 1987; Whyte et al., 1985). Third,having rejected objectivity as an untenable criterion for judging science, a newscientific epistemology was required and has been derived (see Rosser, 1990) byexamining the connections between the discipline and those who use it, and thesociety v.-thin which it develops. This line of reasoning is consistent with a broadrange of thinking in the sociology of science.

The old certainties about science, the old belief in its cultural uniquenessand the old landmarks of sociological interpretation have all gone.(Preface by Michael Mulkay to Brannigan, 1981, p. vii)

Mathematics and mathematics education have been subject to a similarchallenge from within on philosophical, pedagogic and epistemological grounds.The philosophical arguments for a rejection of absolutism in mathematics havebeen explored elsewhere (see Ernest, 1991). Lakatos (1976, 1983), Bloor (1976,1991) and Davis and Hersh (1983) have all made similar philosophical andepistemological criticisms to those outlined in the science literature with respectto the so-called objectivity of mathematics.

Likewise, a critique similar to that found in science has been made ofmathematical pedagogy (see, for example, Burton, 1986, 1990a; Fennema andLeder, 1990; Leder and Sampson, 1989). Despite 'many reports calling forcurriculum reform in nuthematics and science ... the reforms suggested do not

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take feminist concerns into account; in fact, in the case of mathematics they tend

to put added emphasis on curricular areas in which young women regularlyperform less well than their male counterparts' (Damarin, 1991, P. 108).

Mathematics tends to be taught with a heavy reliance upon written texts which

removes its conjectural nature, presenting it as inert information which should notbe questioned. Predominant patterns of teaching focus on the individual learnerand induce competition between learners. Language is pre-digested in the text,assuming that meaning is communicated and is non-negotiable. In Robert Hull's

terms this defines 'knowledge as an object and so equates knowing, and comingto know, with its possession; it effaces the crucial distinction between the learner'ssubjective experience of moving towards knowledge and the objectifying of a

knowledge finally achieved' (Hull, 1985, pp. 49-50).Like science, therefore, mathematics is perceived by many students and some

teachers as 'a body of established knowledge accessible only to a few extraordinary

individuals' (Rosser, 1990, p. 89). Indeed, the supposed 'objectivity' of the

discipline, a cause for questioning and concern by some of those within it, is often

perceived by non-mathematician curriculum theorists as inevitable (see, forexample, Hirst, 1965, 1974, and, for a critique expanding the points being made

here. Kelly, 1986). But 'the processes of knowing (and so also of science) in no way

resemble an impersonal achievement of detached objectivity. They are rootedthroughout ... in personal acts of tacit integration. They are not grounded onexplicit operations of logic. Scientific inquiry is accordingly a dynamic exercise ofthe imagination and is rooted in commitments and beliefs about the nature ofthings' J:lolanyi and Prosch. 1975, p. 63). Adopting an objectivist stance withinmathematical philosophy means accepting that mathematical 'truths' exist and the

purpose of education is to convey them into the heads of the learners. This leadsto conflicts both in the understanding of what constitutes knowing, and of howthat knowing is to be achieved through didactic situations. For example, such

conflicts can be found between the UK mathematics national curriculum,expressed in terms of a hierarchy of mathematical truth statements, and the supportdocumentation given to teachers which includes such relativistic statements as:

Each person's 'map' of the network and of the pathways connectingdifferent mathematical ideas is different, thus people understand mathe-matics in different ways. (Non-Statutory Guidance to the MathematicsNational Curriculum, par. 2.1, p. C1)

The teacher's job is to organise and provide the orts of experienceswhich enable pupils to cnnstruct and develop their own understanding ofmathematics, rather than simply conununicate the ways in which theythemselves understand the subject. (Ibid., par. 2.2, p. C2)

Although 'the ideal of pure objectivity in knowing and in science has beenshown to be a myth' (Polanyi and Prosch, 1975, p. 63), it is a philosophical myth

which continues to exercise enormous power over mathematics both in curricular

and in methodological terms.

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Proposed as an alternative, social constructivism is a philosophical positionwhich emphasizes the interaction between individuals, society and knowledge outof which mathematical meaning is created. It has profound implications forpedagogy. Classroom behaviours, forms of organization, and roles, rights andresponsibilities have to be re-thought in a classroom which places the learner,rather than the knowledge, at the centre. Epistemology too, requires reconsidera-tion from a theoretical position of knowledge as given, as absolute, to a theory ofknowledge, or perhaps better, of knowing, as subjectively contextualized andwithin which meaning is negotiated.

With respect to science, Sue Rosser (1990) has stated:

If science is socially constructed, then attracting a more heterogeneousgroup of scientists would result in difkrent questions being asked,approaches and experimental subjects used, and theories and conclusionsdrawn from the data. (Rosser, 1990, p. 33)

How might including many of those currently outside the nfainstream ofmathematical development influence its conjectures. its methods of enquiry andthe interpretation of its results? In turn, how nUght any changes which resultedfrom a philosophical shiftItiect the pedagogy and epistemology of the discipline?In particular, what are the epistemological questions which are sharpened bybringing a feminist critique to bear on the discipline of mathematics? These arethe focus of this chapter.

Adopting a Cultural View of Mathematics

In writing about mathematics, Sandra Harding (1986) drew attention to its culturaldependency:

Physics and chemistry, mathematics and logic, bear the fingerprints oftheir distinctive cultural creators no less than do anthropology and history.A maximally objective science, natural or social, will be one that includesa self-conscious and critical examination of the relationship between thesocial experience of its creators and kinds of cognitive structures favouredin this inquiry ... whatever the moral and political values and interestsresponsible for wlecting problems. theories, methods, and interpretationsof research, they reappear at the other end of the inquiry as the moral andpolitical universe that science projects as natural and thereby helps tolegitimate. (Harding. 1986, pp. 250-1)

Despite the stance taken by many mathematicians on the objectivity andvalue-free nature of the discipline. David Bloor convincingly argued from a

historical perspective that it is possible to conceive of alternative mathematicsdifkrently derived at different periods:

Seeing how people decide what is inside or outside mathematics is partof the problem confronting the sociology of knowledge, and the

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alternative ways of doing this constitute alternative conceptions ofmathematics. The boundary (between mathematics and meta-mathematics) cannot just be taken for granted in the way that the critics

do. One of the reasons why there appears to be no alternative to ourmathematics is because we routinely disallow it. We push the possibilityaside, rendering it invisible or defining it as error or as non-mathematics.

(Bloor, 1991, pp. 179-80)

More recently, Sandra Harding has pushed the argument further to locatemathematics firmly within its interpretative context despite its overtly comparable

formalistic expression:

There can appear to be no social values in results of research that areexpressed in formal symbols; however, formalization does not guaranteethe absence of social values. For one thing, historians have argued that-thehistory of mathematics and logic is not merely an external history aboutwho discovered what when. They claim that thc gentral social interestsand preoccupations of a culture can appear in the forms of quantificationand logic that its mathematics uses. Distinguished mathematicians have

concluded that the ultimate test of the adequacy of mathematics is apragmatic one: does it work to do what it was intended to do? Moreover,formal statements require interpretation in order to be meaningful ...Without decisions about their referents and meanings, they cannot beused to make predictions, for example, or to stimulate future research.(Harding, 1991, p. 84)

In his discussion of mathematical epistemology, George Joseph (in Nelson et

al., 1993) drew attention to two major philosophical presuppositions whichunderlie western (European) mathematics. These are, first, that mathematics is abody of absolute truths which are, second, argued (or 'proved') within a formal,deductive system. However, he pointed out that dependence upon an axiomat-ically deduced system of proof was a late nineteenth-century development whichwas pre-dated by 'proofs' closer in style to that of non-European mathematicians:

The Indian (or, for that matter, the Chinese) epistemological position onthe nature of mathematics is very different. The aim is not to build up animposing edifice on a few self-evident axioms but to validate a result by

any method, including visual demonstration. (Nelson et al., 1993. p. 9)

George Joseph further stated that:

None of the major schools of Western thought ... gives a satisfactoryaccount of what indeed is the nature of objects (such as numbers) and howthey are related to (other) objects in everyday life. It is an arguable point... that the Indian view of such objects ... may lead to sonic interestinginsights on the nature of mathematical knowledge and its validation.Irrespective of whether this point can be substantiated or not, a morebalanced discussion of different epistemological approaches to mathe-

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Moving Thwards a Fetnittist Epistemology of Mathematics

matics would be invaluable. However, a different insight into some of thefoundational aspects of the subject is hindered by the prevalence of theEurocentric view on the historical development of mathematics. (Ibid.,pp. 11-12)

George Joseph is criticizing the dominance of a Eurocentric (and male)mathematical hegemony which has created a judgmental situation within thediscipline whereby, for example, deciding what constitutes powerful mathematics,or when a proof proves and what form a rigorous argument takes, is dictated andreinforced by those in influential positions. How often do we hear statements,often made about a geometric proof, dismissing it as 'merely a demonstration' orthe suggestion that computer-assisted proofs are not quite as 'good' as thosedeveloped without a computer? How frequently are students encouraged tobelieve that the mathematico-scientific and technological development of theWest has been made independently of a systematic knowledge and resourceexploitation of the rest of the world? The colonization of mathematics has beenso successful that the history of their own mathematical culture and itscontribution to knowledge is often unknown to students in Africa, Asia and LatinAmerica. Such bias is increasingly under attack (see, for example, Joseph, 1991;Needham, 1959; Nelson et al., 1993; van Sertima, 1986; Zaslavsky, 1973) asresearchers uncover the richness and power of mathematical and scientificdeilopment in the non-European world which has been obscured by therewriting of history from a European perspective. If the body of knowledge knownas mathematics can be shown to have been derived in a manner which excludednon-Europeans and their mathematical knowledge, why not conjecwre that theperceived male-ness of mathematics is equally an artthct of its production and itsproducers?

Since I am arguing that mathematics is socio-cultural in nature, the conditionsunder which it is produced are flictors in determining the products. 'Important'mathematical areas are identified, value is accorded to sonic results rather thanothers, decisions are taken on what should or should not be published in a societydetermined by power relationships, one of which is gender. Mathematicalproducts can then be seen as the outcome of the influence of a particular 'reading'of events at a given time/place. Such readings are referred to by Sal Restivo (1992)as 'stories about conmiercial revolutions and mathematical activity, as in Japan, orabout the "mathematics of survival" that is a universal feature of the ancientcivilizations. And they can be stories about how conflict and social change shapeand reflect mathematical developments' (p. 20). In a plenary lecture given at the1994 American Educational Research Association Conference, Jerome Brunerpointed out that explination as causal is a post-nineteenth century phenomenon.A longer history can be tbund for interpretation the objective of which isunderstanding and not explanation. He made out a case for understanding beingviewed as both contextualizing and systematizing and he advocated a route tocontextualizing in a disciplined way through narrative. From this perspective,codified mathematics can be viewed as reified narrative and it no longer seems so

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Leone Burton

absurd to ask how different narratives, or stories, might constitute alternativemathematics (in the plural). Mathematics as a particular form of story about theworld feels, to me, very different from mathematics as a powerful explanation ortool. Re-telling mathematics, both in terms of context and person-ness. wouldconsequently demystify and therefore seem to offer opportunities for greater

inclusivity.

Knowing Science and Mathematics

The feminist literature on the philosophy of science I find very valuable for theclarity with which it has sharpened the critical debate on the nature of knowledge

in science and how that knowledge is derived. However, it is noticeable that thecontent criticisms of science are rooted in the empirical disciplines. For example,female primatologists such as Jane Goodall (1971), Dian Fossey (1983) and Sarah

Hrdy (1986) challenged conceptions of interactive behaviour by refusing to accept

the prevailing (male) views on dominance and hierarchy in sexual selection. Evelyn

Fox Keller (1983) described Barbara McClintock's approach to her study of maize

as highlighting a symbiotic relationship between the plant and its environmentwhich was distinctively different to the more usual 'objective investigation

undertaken by botanists. Rachel Carson (1962) is frequently cited for her earlywork on ecology and the broad view that she took about the environmental effectsof pesticides. In all these cases, the results of the science were different from whathad, formerly, been expected because different questions were asked about what

was being observed and different methods were used to make the observations.However, criticisms of, for example, nuclear physics are more likely to focus

upon the social effects of the science, rather than the science itself (see, forexample, Easlea, 1983). This is not to diminish the importance of developingmodels of scientific "use and abuse which criticize the purposes, products andimplications of scientific developments. But as with mathematics, it is difficult to

confront the abstractions which are the substance and tools of the discipline andthe methods used in their derivation especially where these are analytic and non-observational in order to ask what differences a female perspective would make to

them.In what ways might the questions, or the styles of en(r.!iry or the mathematical

products differ if mathematics were accepted as a socio-cultural construct? Part ofthe difficulty in responding to this question resides in the highly successfulsocialization experiences through which we all go in order to achieve success at

mathenutics. It is exceedingly difficult to dismantle the beliefs which have beenintegral to our learning experiences of mathematics and almost impossible to

construct in our imagirations alternatives to the processes which we have beentaught and with which we have gained 'success'. Hence, scratch a pedagogical orphilosophical constructivist and underneath you are likely to expose an absolutist.

In other words, it might be acceptable to negotiate a curriculum or introduce acollaborative, language-rich environment within which to make the learning of

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mathematics more accessible, but the mathematics itself is considered non-negotiable. However, to be consistent in our critiques, we cannot avoid addressingthe nature of knowing mathematics along with the philosophy and pedagogy ofthe discipline.

Knowing mathematics, and science, has traditionally required entry into acommunity of knowers who accord the status of 'objective', 'in some sense eternaland independent of the flux of history and culture' (Restivo, 1992, p. 3), to theknowledge items as well as to the means by which these items are derived.However, 'objectivity is a variable; it is a function of the generality of socialinterests. Aesthetic and truth motives exist in the realm of ideas, but they aregrounded in individual and social interests ranging from making one's way in theworld (literally, surviving) to exercising control over natural and cultural environ-ments' (Ibid., p. 135). A consequence of this is that 'a mathematical object ... likea hammer or a screwdriver, is conceived, constructed, and put to use through asocial process of collective representation and collective elaboration' (Ibid., p. 137).If we are to argue for a different conception of mathematical knowing from thattraditionally accepted, we must address the meaning which is to be understood by'objectivity' since the truth status accorded to mathematical objects underpins thepervading epistemology. Criticizing scientific 'objectivity' along similar lines,Sandra Harding (1991) has called for an epistemology of the sciences whichrequires a more robust standard for objectivity than that currently in use. Thiswould include the critical examination, within sdentific research (Ibid., author'sitalics, p. 146),

of historical values and interests that may be so shared within the scientificcommunity, so invested in by the very constitution of this or that field ofstudy, that they will not show up as a cultural bias between experimentersor between research communities. (Ibid., 1991, p. 147)

And she further noted that:

the difficulty of providing [such] an analysis in physics or chemistry [and,I would add, mathematics] does not signify that the question is an absurdone for knowledge-seeking in general, or that there are no reasonableanswers for those sciences too. (Ibid., p. 157)

Sue Rosser, in her book, Female-Friendly Science (1990), used women'sexperience of knowing and doing science to draw out differences from what shecalled the conventional androcentric approaches. Amongst many of the possibleinclusionary methods she listed in Chapter 5 are:

expanding the kinds of observations beyond those traditionally carriedout;increasing the numbers of observations and remaining longer in theobservational stage of the scientific method;accepting the personal experience of women as a valid component ofexperimental observation;

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being more likely to undertake research which explores questions of socialconcern than those likely to have applications of direct benefit to themilitary;working within research areas formerly considered unworthy of investiga-tion because of links to devalued areas;formulating hypotheses which focus on gender as an integral part; anddefining investigations holistically

This list, useful as it is for science, does not generalize easily to mathematicsalthough the links to the history philosophy and pedagogy of mathematics aremore obvious. Buchelp appears to be at hand.

Being a Mathematician

In The Emperor's Esi,1 New Afind, Roger Penrose (1990), arguing from the powerfulposition of a research mathematician at the top of his profession, claimed that themathematician's 'consciousness is a necessary ingredient to the comprehension ofthe mathematics. He said:

We must 'see' the truth of a mathematical argument to be convinced ofits validity. This 'seeing' is the very essence of consciousness. It imist bepresent whenever we directly perceive mathematical truth. When weconvince ourselves of the validity of G6del's theorem we not only 'see'it. but by so doing we reveal the very non-algorithmic nature of the'seeing' process itself. (Penrose, 1990. p. 541)

Elsewhere in his book, and in contradiction to the above, Roger Penrosesupported a Platonic approach to mathematics in that he propounded a discoveryrather than an invented, perspective on the discipline. That is, mathematics is outthere waiting to be uncovered rather than within the head (and possibly the heart?)of the mathematiian. And yet, Roger Penrose himself admitted that 'seeing' thevalidity of a mathematical argument must be a personal experience and one which,it seems reasonable to me to assert, can be assumed to differ between individuals.By arguing that 'seeing' is non-algorithmic, Roger Penrose permitted thepersonalization of the process. He reinkirced this with the statement:

There seem to be many different ways in which different people thinkand even in which different mathematicians think about their mathe-matics. (Ibid., p. 552)

However, for ine, far from accepting that the outcomes of mathematical thinkingare discovered mathematical 'truths', the inevitable conclusion Of his statement isthat there are potentially many different mathematics.

The contradiction would appear to lie in a ditThrent perspective on themathematician than on mathematics itself Roger Penrose viewed a mathematicalstatement, once articulated, as being absolute, that is either right or wrongind its

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status verifiable by any interested party But he said, in the

conveying of mathematics, one is not simply communicatingfacts. For astring of (contingent) facts to be communicated from one person toanother, it is necessary that the facts be carefully enunciated by the first,and that the second should take them in individually ... the .fiicnialcontent is small. Mathematical statements are necessary truths ... andeven if the first mathematician's statement represents merely a groping forsuch a necessary truth, it will be that truth itself which gets conveyed tothe second mathematician ... The second's mental images may differ indetail from those of the first, and their verbal description may differ, butthe relevant mathematical idea will have passed between them. (Ibid.,p. 553)

Despite the assumed personal nature of the communication and the expectationof differences in human images and descriptions, there is an assumption that the'mathematics', the essential *truth' of the statement, can and will be the same forall. This is repeatedly refuted by the message of many of the anecdotes which arerecounted by and about, mathematicians. For example, how is it possible tointerpret the kind of intuitive insights which Roger Penrose himself, and othermathematicians such as Poincare, Hadamard, Thom, claim to have had and whichhave led to their finding particular, personal resolutions of a mathematicalproblem. Given that it is reasonable to expect that any one problem might beamenable to a number of different routes for solution, an individual is likely to fallon the one which matches his or her experience, approach, preferences, possiblymaking the mathematical outcome di&rent from that which would be offered byanother individual. Of course, once articulated, the internal consistency of themathematical argument is verifiable. The most recent attempt to prove Fermat'sLast Theorem provided an example of the unverifiability, by most mathematicians,of the claims being made and, consequently, both the potential non-uniquenessand fragility of their status. And, even if the internal consistency is substantiated,this does not additionally encompass any objective status nor any implication ofuniqueness, it seems to me. The social context within which the mathematics isplaced does, however, offer one explanation for apparent uniqueness, or at leastconvergence of *solutions', given that it describes and constrains the 'possible'.Thus, a piece of mathematics is both contributory to, and defined by the contextwithin which it is derived.

A belief in the world of mathematical concepts existing independently ofthose who develop or work with them is attached to embracing the 'objective'truths of mathematics. An image of 'variable' truth, that is degrees of correctness,or solutions responsive to different conditions, is unacceptable to many within thediscipline despite the support ftom the history of mathematics that understandingschange over time as the foci ,md the current state of knowledge change. The socialcontext of a mathematical statement, the impact upon it of the interests, drives andneeds of the person deriving and then conmmnicating it, are disniissed by manymathematicians as inappropriate to the product. Thus, the distinction is made

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between the person who is working at the mathematics, and the mathematicsitself. But I believe that Roger Penrose failed to sustain this distinction particularlyin his discussion of intuition, insight and the aesthetic qualities of mathematicalthinking. He underlined person-ness by reiterating an argument, (see, for example.Thom, 1973, pp. 202-6) that:

the importance of aesthetic criteria applies not only to the instantaneousjudgements of inspiration, but also to the much more frequent judge-ments that we make all the time in mathematical (or scientific) work.Rigorous argument is usually the last step! (Penrose, 1990, p. 545)

In drawing a close analogy between mathematical thought and intuition andinspiration in the arts, he added:

The &balky of inspirational thought is particularly remarkable inMozart's quotation (from Hadamard, 1945) 'It does not come to mesuccessively ... but in its entirety and also in Poincare's 'I did not verifythe idea; I should not have had time'. (Ibid., p. 347)

Any de-personalization of the mathematical process and reification of the productpushes mathematics back into the absolutist position by objectivizing the 'truths'.However, accepting a mathematics which is not absolute, is culturally defined andinfluenced by individual and social differences is not only of great interest to thosewho have argued for an inclusive mathematics but challenges the disciplineepistemologically as well as philosophically and pedagogically.

It does not seem untimely to suggest a theory of knowing that drawsattention to the knower's responsibility for what the knower constructs.(von Glasersfeld. 1990. p. 28)

Once we refocus from knowing that a particular mathematical outcome existsto knowing why that outcome is likely under particular circumstances, we aredistinguishing between the 'objective' knowledge of the outcome and the'subjective' knowing which underlies how to achieve that outcome. This beginsto be familiar as the old debate benveen product and process. However, byattempting to construct a theory of knowing, I am moving past the falsedichotomy of product/process which polarized the how and the what, towards areconceptualization and integration of the how with the what. The value topedagogues of such an approach is obvious. As teachers, we can recognize whena learner mimics a piece of mathematical behaviour rather than acquires it as hisor her own. The articulation of an epistemological position on knowingmathematics which is predicated on mathematical enquiry, rather than receptivimchallenges teacher behaviour. Rather than demanding evidence of the acquisitionof mathematical objects by students, it assumes that mathematical behaviours andthe changes in behaviour that might sign4 learning are products of, andresponsive to, the comnmnity within which the learning is situated. Recountingdifferent narratives, speculating about their similarities and differences, queryingtheir derivations and applications, denies 'objectivity' and reinstates the person and

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the community in the mathematics. Such reconsideration of the characteristics ofscience and mathematics has underpinned much of the feminist work in thephilosophy of science already referenced and is exemplified in the work of SuzanneDamarin. In an article (1991), she presented a table of generalized descriptors 'notas a definitive description of feminist science, but rather as defining a tentativeframework for examining whether and how the teaching ofscience might be mademore consistent with feminist conceptions of science' (Damarin, 1991, p. 112).

As stated above, the philosophical challenge, while not necessarily acceptableto a large number of mathematicians, has been well formulated. (Reference hasalready been made to the work of Bloor. Davis and Hersh, Harding, Lakatos, andRestivo.) Gadamer (1975) added his argument that:

all human understanding is contextual, perspectival, prejudiced, that ishermeneutic [and] fundamentally challenges the conception of science asit has been articulated since the Enlightenment. (cited in Hekman, 1990,p. 107)

as did Elizabeth Fee (1981) to:

attack the objectivity that is part of the 'mythology' of science ... [and]... re-admit the human subject into the production of scientificknowledge. (also cited in Hekman, 1990, p. 130)

Much of the pedagogic challenge is focused on the dysfunction.: :ature of thecontinuum between an absolutist philosophy of mathematics and a transmissivepedagogy and the poverty of the product/process distinction:

On the one hand, authors and publishers produce textbooks that do nothave to be read before doing the exercises; on the other hand, teachersacquiesce by agreeing that this is the way mathematics ought to be taught... the real importance lies not in the students' ability to conceptualize,but rather in their ability to compute. Teachers tend to underscore this bytheir rapt attention to correctness, completenessind procedure. Studentscomply with the grand scheme by establishing as their local goal thecorrect completion of a given assignment and as their global goalreceiving their desired grade in the course. For most, once it's over, it'sover. (Gopen and Smith, 1990, p. 5)

Compare this with a student-centred problem-solving approach:

An instructor should promote and encourage the development for eachindividual within his/her class of a repertoire of powerful mathematicalconstructions for posing, constructing, exploring. solving and justifyingmathematical problems and concepts and should seek to develop instudents the capacity to reflect on and evaluate the quality of theirconstructions. (Confrey, 1990, p. 112)

Researchers have argued that creative mathematicians are more likely todevelop by encountering and learning mathematics in a classroom climate which

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Leone Burton

supports individuals within social groupings; that the negotiation of meaning bothwithin the group and between the group and conventional social understandingsneeds to be encouraged (see for example Davis et al., 1990). These philosophicaland pedagogical critiques, in my view, would be strengthened by the focus,structure and consistency which is gained from an epistemological stance, that is,a formulation of the nature of knowing mathematics.

The Epistemological Challenge

I believe that we can discern the outline of an epistemological challenge tomathematics which, potentially, incorporates approaches consistent with, andfamiliar to, broader constituencies than European, middle-class males. Theseapproaches are inclusive, rather than exclusive, accessible rather than mystifying.encompassing of as wide a range of styles of understanding and doing mathematicsas possible rather thah reducible to those styles currently validated by the powerful.I am claiming that knowing, in mathematics, cannot be differentiated from theknower even though the knowns ultimately become public property and subjectto public interrogation within the mathematical community. Knowing, however:

involves encouraging rebellious spirits to blossom with free rein to theimagination, preserving a certain nimbleness of mind while affording itthe means of being creative. The 'training' procedures, as we conceivethem and ordinarily practise them, hardly lend themselves, one mustadmit, to that kind of enticement, since they more often emphasize thetransmission of acquired knowledge and apprenticeship in proven meth-ods. And considering that those procedures resemble an obstacle coursewhere the competition is tighter and tighter, this hardly encouragesdeparting from the beaten path. (Flato, 1992. p. 75)

I am speculating that five categories, drawn from the work already cited andconsistent with the above critique, might distinguish the ways in which (creative)mathematicians come to know mathematics and that, in their choice ofmathematical areas to pursue. more women (and men) might feel comfortablewith an epistemology of mathematics described in this way. The assumption is thatsuch an epistemology would displace dualisms such as the relativist/absolutistdichotomy and expectations of a value-free mathematics with an hermeneutic andpluralist approach. It would open the way towards an inclusive perspective onmathematics by challenging our understanding of what constitutes mathematics.

I propose defining knowing in mathematics in relation to the following fivecategories derived from the reading reviewed above in the philosophical,pedagogical and feminist literature:

its person- and cultural/social-relatedness;the aesthetics of mathematical thinking it invokes;its nurturing ofintuition and insight;

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its recognition and celebration of different approaches particularly in stylesof thinking; andthe globality of its applications.

Knowing mathematics would, under this definition, be a function of who isclaiming to know, related to which community, how that knowing is presented,what explanations are given for how that knowing was achieved, and theconnections demonstrated between it and other knowings (applications). Whatevidence we have, usually sited in the learning and assessing of school mathematics,suggests that inviting students to define and describe their knowing in mathematic-in these ways does have gender implications (see, for example, Burton,1990b;Forgasz. 1994; Stobart et aL. 1992).

The similarities with Sue Rosser's (1990) and Suzanne Damarin's (1991) listsof the differences between male- and female-friendly science are encouraging. Forexample, both refer to the expansion of the kinds of observations carried out, therecognition of, and concern for, personal responsibility and the consequences ofactions. I have listed a valuing of intuition and insight and the recognition andcelebration of different approaches. Globality, or in both Sue Rosser's and SuzanneDamarin's terms 'holism', is a feature. A need to accept the personal experienceof women as a valid component of experimental observations is acknowledgedwhere I have pointed to person-relatedness which is important to knowingmathematics. Susan Hekman's (1990) analysis of the relationship between genderand post-modernism was also supportive of this approach both in drawing out thesimilarities in argument between feminists and post-modernists as well as pointingout the pervading influence of absolutism in affecting these stances. In HilaryRose's words:

A feminist epistemology ... transcends dichotomies, insists on thescientific validity of the subjective, on the need to unite cognitive andaffective domains; it emphasises holism, harmony, and complexity ratherthan reductionism, domination and linearity. (Rose, 1986, p. 72)

The next step is to open a dialogue with practising mathematicians with aview to discussing the appropriateness of my description to their understanding ofthe nature of knowing in mathematics. This would be done in a style which wouldbe rich in ethnographic data, encouraging the expression of feelings, aesthetics,intuitions, and insights. It would also attempt to challenge the effects ofsocialization into the mathematical culture in order to untangle differences fromcultural similarities. Outcomes which are supportive of the suggested epistemo-logical framework especially where these emphasize impact on gender inclusivitywould provide a strong argument in favour of a re-perception and re-presentationof mathematics. The resulting narrative would have an internal consistency whichshould please all mathematicians.

Such anecdotal approaches AS have already been made confirm the validity ofthe five categories in describing how mathematicians come to know. Argumentsin favour of humanizing and demystifking the mathematics curriculum in schools

22 1

Leone Burton

have long been made with an implication that such attempts change perceptionsof mathematics, and subsequent performance, by formerly underrepresentedgroups. However, these suggestions are rarely connected to epistemologicalframeworks of the discipline more frequently relating either to constructivistphilosophy or empowering pedagogy. And Suzanne Damarin criticizes curricu-lum reformers for their 'reliance on the models of expertise and informationprocessing, which are popular in current research on the cognitive bases ofteaching and learning of science and mathematics [and] appears to be diametricallyopposed to first-order implications of feminist pedagogical research' (Damarin,1991, p. 108).

If the nature of knowing mathematics were to be confirmed as matching thedescription given in this chapter, the scientism and technocentrism whichdominate much thinking in and about mathematics, and constrain manymathematics classrooms, would no longer be sustainable. Mathematics could thenbe re-perceived as humane, responsive, negotiable and creative. One expectedproduct of such a change would be in the constituency of learners who wereattracted to study mathematics but I would also expect changes in the perceptionof what is mathematics and of how mathematics is studied and learned. That sucha possibility, in schools, is not outside the realms ofpossibility is suggested in Boaler(1993). We can also learn from experiences in other disciplines. English, forexample, attracts predominantly female constituencies of learners at the under-graduate level, many of whom have been successful in developing academiccareers. `English was constructed as a liberal humanist discipline which demandedpersonal and thoughtful response ... The most important characteristic of English,in the view of students and staff, is its individualism: the possibility of holdingdifferent views from other people' (Thomas, 1990, p. 173).

Providing a new epistemological context would enable the questioning ofwhat mathematics is taught, how it is learned and assessed within a consistenttreatment. 'By adopting an epistemological view of mathematical knowledge thatstresses change, development, and its social foundations generally, and byconsciously relating this to the curriculum process, the result would be to makethe subject more open in its nature and more easily accessible' (Nickson, 1992,p. 131).

My aim in attempting this work is to question the nature of the discipline insuch a way that the result of such questioning is to open mathematics to theexperience and the influence of members of as many different communities aspossible, thereby. I hope, not only enriching the individuals but also the discipline.

Note

TIUs is a version of a paper first given at the lOWME study group at ICME-7 inQueThec, 1992. Its present content owes much to discussion withind comments from.members of IOWME. In addition, I would like to thank Mary Barnes, Leonie lbws,Stephen Lerman and the anonymous reviewers of ESM for challenging and provoking

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Mm,ing Towards a Feminist Epistemology of Mathematics

reworking of the ideas. This paper is reprinted by permission of Kluwer AcademicPublishers from Educational Studies in Matlwmatics, 28, 3. Special Issue on Gender,edited by Gilah Leder, to appear in Spring 1995.

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)imsi I , J. (1971) In the Shadow of Man, Boston, MA, Houghton Mifflin.GOI'l N, C.D. and Smi III, D.A. (1990) 'What's an assignment like you doing in a course like

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HADAMARD, J. (1945) Tlw Psychology of Invention in du Mathematical Field. Princeton. NJ,University Press.

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Chapter 25

Mathematics: An Abstracted Discourse

BettyJohnston

It is clear that we wouldn't have written this book and you wouldn't be readingit if we weren't all concerned in sonie way about 'the gender imbalance inmathematics'. What is not so clear is just how we understand this imbalance, whatwe believe its causes are, and what we think we shbuld do about it.

I have questions about this imbalance, which I cannot address here. How doesthe imbalance manifest itself? What 'facts' are we using to help us see it, whocollected them, for what purpose, on what evidence? What does 'good at maths'mean and how do we measure it? How do we construct our understanding of the'facts'? How do we use it? And, finally, why do we care so very much that everyoneshould do mathematics? These are all questions addressed elsewhere by peoplesuch as Yves Chevallard (1989), Richard Noss (1991), Dorothy Smith (1990), andValerie Wilkerdine (1988).

All sorts of reasons have been proposed to account for the indisputable,widespread fear and dislike of mathematics. Some of these reasons are specificallyrelated to women, others are more generally applicable to groups in our societywho miss out all are variations, more or less sophisticated, on the theme 'youmust have been away at some point', or 'you must have had a bad teacher', or'perhaps you thought it was unfeminine', or 'maybe you haven't got a mathemat-ical mind'. And while all this weaves into the complex pattern that is ourexperience, perhaps we don't take seriously enough the voices that say, again andagain, 'but it doesn't make sense', and 'what's the point of it?' Perhaps what theyare saying simply is true. Perhaps mathematics, their mathematics, secondary-school mathematics, doesn't make sense. Perhaps the fault is in the mathematics,and not the teaching, not the learning, not the people. At the very least, it is aquestion worth focusing on for a while.

Mathematics is a human construction and learning mathematics is a socialprocess (Ernest, 1991). Western society is saturated with numbers. Quantification,rationality and abstraction arc intricately connected, increasingly embedded andhighly valued in the scientific and technological world. What I am interested inis the interaction between mathematics and society, in how, for instance, thequantification of society constructs us particularly, as women, and in how we

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construct or resist it.. As Sue Willis (1989) argues, attempts to explain the genderimbalance shift the focus from girls can't (they are biologically incapable of doingso), to they don't (they are not socialized into that particular role), to they won't(they choose not to). The emphasis changes first from biological determinism toa more complex social determinism, and, from this still quit: passive picture ofwomen, to one that insists that individuals are active participants in their formationas mathematical beings.

Two theorists who have grappled with the question of agency within structureare the Canadian sociologist, Dorothy Smith, and the German sociologist andpsychologist, Frigga Haug. Both work from within feminist and socialistframeworks; both begin from, and value, the standpoint of women; both developmethodologies that allow examination of the interface between the everydayworld and the wider social structures. Both approaches would be valuable tools forour research, but I will focus here on the use of Haug's memory-work.

In examining the process of female sexualization, Haug (1986) developed amethodology that has become known as memory-work. In the last five years,variations of memory-work have become quite widely used in a number ofresearch projects worldwide. In Haug's original use of the method, a group ofpeople interested in a given topic, and preferably from a variety of backgrounds,jobs, and disciplines, meet regularly. They write and analyse relevant 'memories'from their own lives; they collect and examine other materials documents bothold and new, dogmas, fairy tales, proverbs, newspaper articles; they examine anddiscuss theories and opinions about the process they are focusing on. They do allthis from a theoretical viewpoint that seeks to use experience as the basis ofknowledge and refuses to view people simply as bearers of roles, but sees them asactive participants even perhaps in their own subordination. Gender, like class andethnicity, is neither chosen nor given, but produced and therefore transformable.Memory-work differs from autobiography, being not an investigation of the waythings really were with continuities retrospectively constructed, but a study of theprocesses through which we have become the people we are today. Haug andothers recognize that individuals do not give objective accounts of themselves.But, they say, our reinterpreting, falsifying, forgetting and repressing do not needto be seen as obstacles to the truth, rather they can become opportunities forinvestigation. In examining how individuals construct their identity, and whatbecomes subjectively significant. we are examining how women grow into thestrucwres of society (Haug, 1986, p. 40). The emphasis of the investigation is noton controlling others, but on taking control for ourselves.

I also wanted to examine the action of people within given social structures.to try to 'identify points at which change is possible, where a shift fromhewronomy (subjection to external laws) to autonomy can take place' (Haug,1986, p. 41). To do this I have used the process of memory-work with a group ofwonwn in Sydney to reconstruct the process of mathematization through anexamination of our own past experiences. We tried to tease out the ways in whichthe discourse of mathematics, all the varied knowledge, practices and beliefssurrounding mathematics, works to insert us into its world, whether as successes

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Betty Johnston

or not, and, how we, as individuals, weave ourselves into the mathematical world.We met for two or three hours every two weeks, usually on a Sunday morning,over a late breakfast, for a period of four months. The women involved in thegroup were, as in Haug's original group, from a variety of educational and workbackgrounds a high-school science teacher, an adult-literacy teacher, a

librarian, an engineer, a design student, a numeracy lecturer, a research assistant.This variety brings a range of experience and theoretical frameworks to thediscussions and analysis, but we were not randomly chosen and so in no sensecould we be seen as 'representative' of our particular backgrounds. Like Haug(1986), we are not using memory-work to make statistically valid statements, somuch as to explore possible relations and dynamics, and possible transformations.

The memories and our discussions have revealed a wilderness of startingpoints: an absence of mothers and a persistence of fathers; a felt connectionbetween sin and correctness; a surprising sensuousness associated with many earlymemories of doing mathematics; widely differing understandings of the meaningof 'cleverness'; mathematics as order, and order shifting between pattern, sequenceand command; the violence of measurement; n,:asurement and accountability. ...In this chapter I trace one strand that makes connections between genderimbalance and the discipline of mathematics.

Listen to three of the stories written in response to the cue: 'doing maths'.Members of the group were asked to write about an early experience of doingmathematics, and to write it in the third person in an effort to distance themselvesand to prevent them censoring the stories, leaving out what might be significantconnections.

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Kat.Her memories are more impressions: maths was part of the fluid world ofbeing in a cocoon which was school impressions of warmth, laughter,friends, the certaility of the sums and the rules. Chaos was not a part of diisworld. Order, cut grass, humming summer heat and security.

Alison8.30 am. Convent school under a tree. The three girls are animatedlydiscusiing their maths homework from the previous night. Not set home-work. This was stuff they'd decided to do, they couldn't wait to do, a well-worn mathematics pathway that had been trodden by thousands before them,a journey of discovery ... The usual hum of maths conversations for thesethree students. Lynne could not get out problem 56, a tortuous piece ofgeometry which had left a smudgy trail of carbon pencil over the page. Thesame page was already indented by the pressure of earlier problem,s, so artfullydeciphered that their presence could be felt: as your hand progressed across thepage. What a turn on! Sheila had fathomed 56, and proudly unveiled hersolution ... Lynne can't believe she didn't think up the solution herselfshit! With a flurry of rubbers, Lynne's page is erased. The now famoussolution is recorded. The page is now even more smudged and creasedbeautiful record of lunnan endurance and application. It looks fantastid This


is the best part what visuals! Their own shared mathematical aestheticism.

MarieShe's about 11-years old, crouched over her work, arms around the outskirtsof her books, hiding it. Palms sweating. The extent of her shame, notknowing, dirtiness, must somehow be contained and hidden. Then there is afigure behind her, towering, menacing: Sister Peter. Her voice is cold and lowand in a few words the shame, not knowing, dirtiness of the young girl areexposed, public. 'That's wrong. What do you think you're doing. Tear it outand start again.' Repeat the shame, repeat the humiliation.

What links were being made? What silences were apparent? One link thatsurprised us was the strong sensuousness of the stories, from Kate's ease andsummer, to Alison's crumpled paper and messy pencil which she later describedas the physical representation of thinking, to Marie's connection of failure withmess, dirt and towering authorities. A silence that we noticed, after some time, wasthat none of the stories actually contained any mathematics. It seemed that notonly had we learnt in context, we had learnt a context: it was the whole contextthat was remembered, though, except for Alison who had retained a little, themathematics had fallen out. Why was the mathematics itselflost?

An Ordered Daily Training ...

The point I want to follow here is our realization that the stories seemed to be notdescriptions of a single occasion so much as a condensation of a number ofrepeated similar experiences, layered memories. Marie's memory, for instance,reflected the continuing, everyday pettiness and horror of her first year at high-school; what we are seeing in each story is an ordered daily training towards aparticular mathematical normality.

... By the Reduction to Mathematics of the Everyday World

Much of the mathematics that most people use routinely is concerned withmeasurement, with the breaking up of wholes into parts. There is hardly an aspectof our lives free from this quantification. The history of the measurement of time,for instance, demonstrates its increasing use as a means of controlling workingpopulations (some schools in New South Wales have teaching periods of 52.5minutes!). Are there differences in how men and women experience thisquantification? Why, as Alison asked, did boys watches have a second hand, whilegirls' only had hours and minutes? One strand in the struggle to be an activeparticipant in her own life emerges for Marie from the unravelling of hermemories about time, and connections between paid work and the measurementof time:

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Part of the struggle for me around work has been that I feel like I neverdo a good enough job and I'd be much happier working for nothing,because I can be myself, and in fact my experiences with teachingabsolutely confirm that ... that when late last year, I worked withsomebody for nothing, well for an exchange of energy she's going todo work on my garden it was great, but when I work for money inmy job full-time I ... feel totally distressed, all the time, ... and a feelingthat I have a lot at work is that my time isn't my own ... there's all thisfunding pressure now to take particular kind of people into our groupswho will be able to articulate into the next exciting episode of TAFE, insix months time ... it has been fairly open-ended, it depends on people'sgoals, but now it's well a thing of people can't spend as much time ... butfor a lot of people, yeah, that's what they are wanting, that sort of controlover their own learning and ... confirmation and stuff, but it actuallytakes time to do that.

Women's work, paid and unpaid, is often described as a 'double burden'.Frigga Haug (1992, p. 260) argues that it is more accurate to say that 'women arelocated in two areas with contradictory logics of time', the measured time of paidwork and the unpaid, and therefore unvalued, situations where spending moretime is better than rationalization. And so, she suggests, a certain resistance tomathematical thinking may be part of women's upbringing in order to preventschizophrenia.

... By the Separation of Mathematics from the Everyday World

For Kate also, the normality constructed in the process of mathematization,involved a not-belonging, a separation of thought and experience.

2.30

So maths geniuses are young because ... because maths is so ... thin, andthey can say: I can understand it, because they think they only need tounderstand that much. I mean they are just totally unaware of what theydon't understand ... there are people who are very good at spottingpatterns, and then there are other people, but they are very rare, whowhen they look at a .page they can see a pattern and it actually has ameaning in terms of reality for them ... the only way I can describe it isit has a 3-D reality, it's a real thing that's happening, but they can also seea pattern ... and I personally think the people who are not good patternspotters are people who right from an early age have demanded that levelof understanding, rather than it being the other way round ... like I'vebeen able to do maths because I've accepted that I shouldn't ask questions... I knew that I had pattern-spotting skills but suddenly they weren'tworking and so why weren't they working and it linked into this wholething about not having this history of 3-D objects and in practical

k


situations linking with patterns ... I mean a lot of the guys haven't madethe connections between their experiences and pattern making but theydo during the course ... suddenly the course is making sense of theirreality.. ..

Like Kate, none of the women in the group had had the experience of thesemen in the engineering course, of mathematics reinforcing their reality.

... In the Normality of Heteronomy

As Marie's early stories unravelled, at times very painfully, we often recognizedourselves. 'With maths, everything had to be laid out and ordered and mine justnever was ... and also you knew there was a right answer ... that I couldn'tnecessarily arrive at through my own process ... the right and wrong dichotomyis so strong and then . . there's a link with sin.'

Her sense of the external structures defining her responses, of herself as avictim, continued into high-school.

I see school as a quite profound violation ... I think I'm a very skilledmimic and what started out as a way of surviving, picking up what wassocially accepted behaviour and just reproducing it ... while leavingsomething intact inside ... I think I actually really lost myself ... Mymemories of maths at school, was that it was a construction that otherpeople made for me and of me, but that I never shared myself with ...and because I feel like I was mimicking and that things were not comingfrom within, then I have always had a very deep feeling of being foundout one day ... that one day the axe would fall down ... and I actuallyjust remembered today that I did all right in mathematics, like I got anA for School Certificate [fourth year of high-school], and thinking, yesI got away with that, but I might not be able to sustain the act.

To learn every day that it is normal that mathematical knowledge is externallygiven and monitored, that patterns reflect no reality, that a quest for a certain kindof understanding hinders success, that everyday practices are quantified andregulated by a vast array of indices, is to experience mathematics as a profoundlydecontextualized discourse: an abstracted discourse that could refer to anything,and for most people refers to nothing (Walkerdine, 1992). It is to experiencemathematization as the ordered daily training in the normality of heteronomy.

The abstract mathematics that most of us know, like economics, ignores bothits sources and its applications and consequences. That industrialized societyshould have such decontextualized mathematics is no accident, Peter Denny(1986), in a study that compares the mathematics of a hunter society, the Ojibwayand Inuit peoples of North America, with that of agricultural and industrialsocieties, has argued:

[Our content-free] style of mathematics mirrors the style that is required

23 1

Bettyphuston

by industrial technology. ... the isolation of crucial variables and theirmanipulation independent of context is needed for the high degree ofalteration of the environment achieved by industrial society ... It is

inappropriate however for people in hunting societies who earn theirliving from a relatively unaltered environment ... Since [the hunter] willnot be controlling nature, isolated knowledge of crucial variables will nothelp, but inclusive knowledge of the whole pattern of natural processeswill be imperative. (Denny, 1986, p. 142)

The difference between the two kinds of mathematics is clear in the case ofnavigation. The navigator in industrial society can derive a direction from a singleselected factor, magnetic north, and maintain this information in isolation by usinga gyroscopic compass even if the magnetic field becomes disturbed. The navigatorin a hunting society must pay attention to dozens of factors simultaneously. Thelimit hunter travelling by dog-sled through a blizzard must register changingconditions of snow, ice, wind, temperature, and humidity, all in large complexpatterns, which, taken as whole structures, will successfully specify location. Thishigh degree of contextualization can be achieved only if there is a general habitof treating information in context rather than in isolation. Therefore it will notsurprise us if this inclusive style of thought shows up in the mathematics of huntingsocieties, just as the isolating style shows up in the mathematics of industrialsociety.

In the industrialized world, mathematics has become so 'content-free' that, forexample, the metric system gives as standard measures of length, millimetres,centimetres and metres the first two, too small, and the last, too large forcomfortable measurement of many everyday objects. Feet and inches may havebeen less elegant, or less useful to science, but a few inches or a few feet effectivelymeasured most things that ordinary people needed.

In modern, industrialized societies, with their proliferating global problems,increasingly complex and interrelated, such isolation is extremely dangerous andirresponsible. Abstract reasoning, Valerie Walkerdine (1992 ) argues. is not theultimate pinnacle of intellectual achievement, but a massive forgetting which isintricately bound up with questions of power. Perhaps we should bear in mindHegel's warning:

[the analytical approach] labours under a delusion if it supposes that whileanalyzing the objects, it leaves them as they were: it really transforms theconcrete into an abstract. And as a consequence of this change the livingthing is killed: life can only exist in the concrete and one. Not that wecan do without this division, if it is our intention to comprehend ... Theerror lies in forgetting that this is only one half of the process, and thatthe main point is the reunion of what has been parted. (Hegel, 1975,p. 63)

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h. 1 .1

Mathematics: An Abstraded Discourse

From Heteronomy to Autonomy

If by 'discipline' we mean 'a branch of instruction or learning', then when weblame the discipline of mathematics itsclf for the gender imbalance, we are guiltyyet again, in our essentially mathematical habit, of isolating a single variable andtreating it out of context. We are guilty of inappropriate abstraction. To return itto its context we need to consider not only its knowledge, but its practices, itsconstruction of meanings and emotions, and its constitution as a process thatinserts us into given social worlds.

A step towards autonomy would be an appreciation of mathematics, not as themanipulation of the already abstracted, free-floating products of abstraction, but asthe process of abstracting from the material context, and not only that, but ofreconnecting to the concrete, to the context, as we are learning to do in ecology,learning to embed our knowing in an understanding of sources and consequences.

It was not until recently that Marie took a course about adult learning:

... that's the only course that I've done that has in any sense come fromme and touched me and it marked quite a profound personal shift ... myfirst experience of that, of knowing in general and quite specifically inmaths was with the maths anxiety workshop and the handshake problem... and this whole revelation, that it has a basis in concrete experience,that is if I understand the reality of it then I don't need to know the rulesbecause I can just get there again .. I could get back to the formula.

And if you have people coining from their own experience, you havepeople who have self-confidence and are not fooled by Hewson and therest of the politicians ... and that's something I experienced in this course... this extraordinary sense of being in a process, that everything hasmeaning ... and could be, like we're doing here ... really enjoying usingour own experience, no not usi,lg it, talking about it, valuing it to workon it, bringing together some ideas, to develop some knowledge .. it isreally exciting and sort of passionate and enjoyable, but we have peoplewho are not socially controlled I think, in the end ... and who come frombeing very damaged to increasingly feeling, I am worthwhile.

And so we travelled with Marie from a position as victim, from her trainingin the normality of heteronomy, to the possibility of ourselves also as actors, to thepossibility of autonomy.

And What Are the Costs?

Even more fundamentally, maybe we should be asking a different question, notwhy is there an imbalance, why don't girls do as well as boys, but why does anyoneever do well in this 'thin' subject (as Kate called it)? And what are the costs,individually and socially of the ordered daily training in the normality ofheteronomy?

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References

CI ILyAl I Atm Y. (1989) 'Implicit mathematics: Its impact on societal needs and demands',in MAL ONE, J., BURKILARI EE, H. and KEEFE! , C. (Eds) The Mathematics Curriculum:Thwards the Year 2000, Perth, Science and Mathematics Education Centre, CurtinUniversity, pp. 49-57.

Di NNY, J.R (1986) 'Cultural ecology of mathematics: Ojihway and Inuit hunters', inCi oss, M.R (Ed) Native American lathematks, Austin, University of Texas Press.pp 129-80.

ERNES1, P. (1991) Tlw Philosophy qf Mathematics Education, London, Falmer Press.HAUL, F. (Ed) (1986) Female Sexualization: A Colleaive Work qf Memory, translated into

English by Erica Carter, London, Verso.HALA, F. (1992) Beyond Female Masochism: Memory- I4,'ork and Politks, London, Verso.HI (di G.WF. (1975) Logic (Encyclopaedia of the Philosophical Sthlkes, Part 1), translated by

W Wallace (3rd ed.). Oxford, Clarendon Press, Section 38z.Noss, R. (1991) 'The social shaping of computing in mathematics education', in Pirvim,

13 and Lovt, E. (Eds) 77w 7i.achim and Learning tf School Mathematics, London, Hodderand Stoughton, pp. 205-20.

SMI I II, D. (1990) The Conceptual Praaices Power: A Feminist Sociology qf Liowledge, Boston,MA, Northeastern University Press.

W IM)INE, V. (1988) The Atastery of Reason: C:ognitire Devehpment and the Prodmtion qfRationality, London, Routledge.

WAI kl ROINI.. V. (1992) 'Reasoning in a post-modern age'. Paper presented at the 5thInternational Conference on Thinking. Townsville.

Wil is, S. (1989) 'Real Girls Don't do Maths': Gendo and the CoIntruction (!). Prieihge,

Geelong, Deakin University Press.

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Chapter 26

Constraints on Girls' Actions inMathematics Education

Marjo lijn Witte

Introduction

The Dutch education system is based on a view that sees education as thetransmission of knowledge, rather than as supporting the construction ofintegrated personalities, starting from individual differences. This emphasis onknowledge transmission creates a mathematics education that encourages studentsto model themselves after the experts who created mathematics. In this chapter,I argue that this is a view that also inhibits the participation of girls inmathematics.

Many theories have been advanced to explain why more girls than boys inwestern society drop out of mathematics education (Sherman, 1981; Pedro et al.,1981; Leder, 1985). And, in spite of various attempts to change this situation, thispattern still persists (see, for instance, Sangster, 1988; Burton, 1990). Of course, themere fact that girls prefer careers that avoid mathematics need not constitute aproblem in itself provided it is a choice. Given this, the question is, 'On whatlevel, and what kinds of, intervention will be most effective in remedying thesituation?'

Three Intervention Strategies

In this section, three different change strategies are dis.,:inguished. Most inter-ventions fall into the tirst category which emphasizes individual choice and factorsthat determine this choice. The second category focuses on the educationalenvironment, while the third conceives of mathematks education as a subsystemof society. influenced by a dominant language. This language puts constraints onthe actions of learners and, as I will argue, disadvantages girls.

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Improving the Student

The most frequently employed intervention strategy operates on the level of theindividual girl. This strategy is based on hypotheses about what the average girl

wants or can do. It may emphasize biological differences, such as girls having lowerspatial-visualization abilities than boys, or the social construction of femininity as

ways of behaving, choosing and acting. The aim of this strategy is to look forfactors that might determine high gender-related drop-out rates and negativeattitudes towards mathematics education, and use these factors to inform the

design and implementation of interventions.An example of this approach in the Netherlands is the governmental

campaign, 'Choose Exact', which tries to influence girls' choices towards themathematical sciences. According to many theories about girls' attitudes, girls lacka proper awareness of the usefulness of mathematics. This diagnosis has beentranslated into the notion that girls have to be convinced that well-paid jobs aremore accessible if they choose to study the exact sciences. The campaign failed toreach its target. Certainly, girls became more aware of the relevance of the exactsciences, but this awareness did not change their choices (Ministerie vanOnderwijs en Wetenschappen, 1990). Probably usefulness does not simplytranslate into career possibilities.

A more essential criticism of this approach is that it is too ambitious. Thefactors that seem to influence girls' choices are so many and so varied, and sointerrelated, that choosing any collection of them on which to base an effectiveintervention is rather like buying a ticket in a lottery A second drawback with this

approach is that it focuses only on the girls and what shapes their behaviour ingeneral. Very little attention is paid to their contact with mathematics as adiscipline, as mediated by the educational system (for a feminist critique of themale character of much of science and mathematics see, for example. Keller, 1985;Harding, 1986; Walkerdine, 1989).

Improving the Em,ironment

The basic idea behind the strategies in this category is to improve the educationalenvironment in order to attract girls. This strategy as applied in Dutch secondarymathematics education, has produced many changes in the curriculum and inmethods of teaching. The most recent change was aimed at developing a form ofmathematics education which was more open to all learners, and supporting themto the maximum of their abilities and interests. Any remainMg diffin.ences, it wasargued, would therefore be due solely to invariant differences at the individuallevel. This development, called 'realistic mathematics'. deemed it important tobuild upon what pupils already know, their real-life experiences, the variousconcepts and procedures they have already developed. Hence, many differentindividual conceptions of 'mathematics we assumed possible. However, in theinitial trials, 'realistic mathematics' was not as successful as hoped. Boys were moreattracted by the contexts chosen than were the girls whose interests and

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Constraints on Girls' Actions in Mathematics Education

motivations were not encouraged as much as had been hoped (Dekker et al., 1985;Vos, 1988). These so-called 'realistic contexts' seem not to appeal to girls' reality,and any attempt to make them more attractive to girls cannot succeed as long asonly one kind of reality, the kind that can be (ill-)defined by mathematical means,and not reality as individuals see it, is permitted in mathematics education (Van derBlij, not dated). Very striking, in this regard, is de Lange's (1985) finding that itis often girls who point out the limitations of a mathematical model. Determiningintervening factors in order to improve educational practice appears to be difficult.

Improving the System

In this approach, the content of mathematics education, or mathematics as a schoolsubject in the context of society, is examined. Mathematics is not viewed ascomplete and immutable, but as a field which changes over time. Preferences inmathematics curricula relate to distinctions in the job market; distinctions in thejob market relate to behavioural differences; behavioural differences relate tolinguistic differences, etc. These relations are important for they tend to increasethe stability of sockty.

In the remainder of this chapter I will focus on the third strategy outlinedabove. In the long-run, improvements at the level of the system, rather than at thelevel of the individual or the environment, may prove to be more effective inachieving a more permanent change in gender-related preferences for, andbehaviour in, mathematics education.

The Language of Mathematics Education

In my view, mathematics education in the Netherlands, and in many othercountries, is based on a certain dominant conception of mathematics: although thenotion of mathematics as an open, increasing system seems to be accepted intheory, in practice mathematics itself is not a subject which is open to debate. Insecondary school mathematics, th objectives are strictly prescribed, as ifmathematics were just a set of facts that children can acquire or re-invent and usein different domains. 'Realistic contexts' are used as a means of rebuilding a pre-existing mathematical system. For example, 'the rules for differentiation ofelementary functions are "discovered- ,r "re-invented- before the formal rules arcfinally given and proved ... Finally, the students [embarki on the formal-analyticalapproach. The process of conceptual mathematization ends up with the differ-entiation rulesind their formal proof's (de Lange, 1987, pp. 71-2).

I will distinguish between two basic conceptions of mathematics. Thedonnnant conception focuses on closed fbrms of mathematics, the validity ofwhich is based on assumptions or axioms that need not be justified by localcontexts. For this reason I will call it 'expert-mathematics'. Inspired by the ideasof the Dutch mathematician, Brouwer (1907), the second conception of'mathematics focuses on the design of various forms of mathematics which are

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Madoliin Witte

justified by local needs and conditions. I call this second conception ofmathematics 'user-mathematics'.

Expert-mathematics

Expert-mathematics is based on an absolutist view of knowledge which pre-supposes that we can distinguish objects in our world and represent them byabstractions. The relations between abstractions are described, isomorphic to themore 'concrete' relationships between the objects. Abstract relations are expressedas sentences in a formal language and constraints are put on which sentences,expressed as axioms, are relevant. Basing itself on abstraction, as it does, thislanguage structures people's activities in this environment by focusing onmeasuring, computing and comparing.

Lakatos (1978) describes Euclidean geometry as the deduction of knowledgefrom trivial, true propositions (or axioms), on which the mathematical system isbuilt. The deductive, or axiomatic, method is used to decide which sentences orpropositions are accepted as belonging to the mathematical system. The legitimacyof the axiomatic method is beyond the realm of control of the user of mathematics,and lies instead in the presupposition of the universal knowledge of nature, seenas an external authority. It is assumed that it is possible to make statements aboutreality by basing a mathematical system on axioms which stand to reason, andcircumscribing the possible theorems by quality criteria such as coherence,consistency and completeness. As Lakatos (1977) says, 'in deductivist style, ...[miathematics is presented as an ever-increasing set of eternal, immutable truths.'

The properties of 'expert-mathematics' guarantee 'a steadily accumulatedbody of knowledge, linear, hierarchical, dependable, reliable and value-free.Concepts do not develop, they are discovered' (Lerman, 1986, p. 71). 'Counterexamples, refutations, criticism cannot possibly enter' (Lerman, 1986, p. 142).

Tlu. Restrictions on Pupils' Actions

Expert-mathematics implies a certain organization of the curriculum, which putsclear restrictions on pupils' actions, both in terms of what they are allowed toproduce and what processes they are allowed to use to reach the desired results.Not all of these are necessary, and therefore, as Lakatos states, 'it has not yet beensufficiently realized that present mathematical and scientific education is a hot bedof authoritarianism and the worst enemy of independent and critical thought'(Lakatos, 1977, p. 142). As I stated above, the emphasis on knowledge-transmission creates a system of mathematics education in which students aretaught to mimic experts. Children learn knowledge which is not necessarily usefulto them in their (personal, or local) situations, or for solvMg their own problems.They are under the control of the teacher or the school textbook, outsideauthorities in matters which they feel are also their own, and this can lead tofeelings of stress. Confrey (1985) argues that if rules, procedures and mathematicsare seen as steady certain and objective, children will perceive their ow,1 ideas,

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procedures, constructions and reflections as inferior and irrelevant. According toConfrey, this results in a negative attitude towards the subject, and generates apathyand aversion.

The dominance of expert-mathematics is also visible ii? Dutch realisticmathematics education. Although, in theory, children have freedom to explorestructures in real-life situations, in practice they are allowed to find only prescribedstructures in situations presented by their textbooks. Situations are chosen byeducators because they provide students with suitable contexts for acquiringcertain prescribed knowledge, rather than leaving children to construct or developtheir own knowledge through exploration within the given context. Theproblems which are posed have implicit boundaries, which structure the possiblesolutions and the permitted activities. Because expert-mathematics is mainly usedin science or technical domains, the contexts and problems which are chosen ineducation are mostly from those areas. Students whose 'reality' is not such atechnical one and who respond in these contexts by avoiding expert-mathematicsare viewed as mathematically untalented. The majority of these students arefemale. The school system, as it is presently organized, apparently inducesavoidance of mathematics by women. To remedy this situation, we must abandonexpert-mathematics and absolutist knowledge theory

The Possibilities of User-Mathematics

User-mathematics, the second conception of mathematics mentioned above,provides an approach to mathematics education that holds much promise forincreasing the participation of women in mathematics. It is related to a

constructivist view of knowledge, the main proponent of which is Brouwer(1907). Mathematics, in Brouwer's view does not deal with the representation ofobjects in reality. It must be seen as the product of human activity, as a mentalconstruction, possibly shared by more than one person. According to Brouwer, thebasic mathematical operation is the indication of a sequence of events in time. Itis also the basis of action. 'These sequences were of a new kind: they were' notgiven by a law (the automatic guarantee of infinite extension!) but by a choiceprocess of the creating mathematician' (van Dalen, 1991, p. 9). Each person canmake choices in a difrerent way, and can order things in a way that is dependentof the needs and preferences of the individual. In this way, each person builds hisor her own mathematical system. The main characteristic of user-mathematics isthe legitimacy of the mathematics constructed. This is not necessarily bound to acertain 'objective reality', but may be subjected to .1 person's own needs andprefin.ences in a particular situation. It is not necessary to stress external qualitycriteria, such as logical consistency or completeness. Every ordered series ofevents, produced by an individual, is seen as a certain mathematics, and theproduced order is added to the series. In this way, external legitimacy is replacedby internal legitimacy. There is no need for any cornsspondence between relationsand objects in the environment or in reality 'I do not recognize as true, hence as

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Alariolifit Witte

mathematics, everything that can be written down in symbols according to certainrules, and conversely I can conceive mathematical truth which can never be fixeddown in any system of formulas' (Brouwer et al.. 1937, p. 452). 'In this(Brouwerian) sense, mathematics is the overall systematic discipline dealinglegitimately with everything (van Dalen, 1991, p. 10). Mathematics gives expres-sion to what people do, not to what they expect from their environment. Lakatos(1977) represents the same view when he states that mathematical activityproduces mathematics. With a certain autonomy, mathematics develops as a living,growing organism from the initial activity. It emphasizes the way individuals dealwith their environment, using its properties as constraints.

(7ser-1iathernatics in Education

User-mathematics construes a different type of pedagogy from the one requiredfor expert-mathematics. In expert-mathematics, children are bound to see theregularities that teachers or curriculum designers have designated for them to see.In user-mathematics, by contrast, children have to find regularities in their ownpersonal environment. They act as creative subjects; their activities are directedtowards the preservation and extension of their own universes, rather than towardsthe design of an 'objective' universe. User-mathematics does not oblige studentsto use the same model, which is also not necessarily bound to a certain calculus.The users do not have to be identical, as expert-mathematics requires. User-mathematics does not select students on the basis of the possession of certaintalents, or on the ability to solve calculable problems. Education of this kind is notrestricted only to a slect group of students, who recognize the prescribed world-view

Several concerns about user-mathenutics should be addressed. First, providingtbr more subjective constructions does not necessarily make pupils less effective indealing with the world. Such constructions also take in what children want to do.

Second, although children may impose linguistic constraints in their constructionof mathematics, there is still the possibility of their communicating and sharingcertain language elements. Children will be 'constructing their own knowledge,by comparing a new problem, idea, object, hypothesis against their existingexperiencemd conceptual system' (Lerman, 1986. p. 73). Shared constructs andshared knowledge are possible, but arc not necessary. Both, however will fulfil therole of coordinating principles, rather than fiinction as descriptions of what is.

Third. there is a serious danger of confusion in going only half-way. Whenteachers decide to adopt an approach which allows for the possibility of inurepersonal and individual constructions, they often make problem-solving andproblem-formulation their main forms of expression. The language of problem-solving then begins to dominate. All of n mathematics is retbrmulated in its terms.but the underlying expert-mnatheinatical nature of knmyledge is maintained. 'Flueresult is a confusion between what teachers take ati a constructivist approach. andwhat they actually do.

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COM .4CtIon.s III A lathematio Ethication

User-Mathematics: A Promise fi)r Girls?

In support of my claim, that user-mathematics will increase the participation ofwomen in mathematics, I refer the reader to two situations in which students wereable to choose their own methods of problem-solving using LOGO. In both cases,the authors. Sutherland and Hoyles (1988) and Turkic (1984). concluded thatgender was no longer a reliable indicator of student motivation and pertOrmance.It is impossible for me to claim that every girl will have more affinity formathematics after engaging in user-mathematics education. Neither can I claimthat more equal numbers of boys and girls will choose to participate in moreadvanced mathematics, or in mathematically related domains, especially if thesedomains restrict themselves to current forms of mathematical modelling. But thereis evidence that the choice and the exploratory mode that is characteristic of user-mathematics, and seldom present in expert-mathematics, is more attractive to girls.Apparently this is so, even in the area of assessment, for 'The creative constructiveaspect of the take-home test, or the two-stage test (or essay), seems to especiallyappeal to girls. Most of the very best results came from girls' (de Lange, 1987,p. 261).

Examples of User-Mathematics in Education

There are few, if aiw, examples of the full implementation of user-mathematics ineducation. But there are examples which meet some of the criteria. These criteriaare two-fold: first, the user is stimulated to a (personal) production of events; andsecond, users put those events in a certain order, including the order which hasbeen produced. in order to deal with events and be more effective in the situation.To clarify this. I give two examples.

In a Dutch mathematics school textbook, a task is presented about holidays.In this (expert-mathematics) task, the students have to compare the costs of rentingand driving two different cars over different distances. They are encouraged to beas clever as they can, but they are only allowed to operate within the very restrictedarea of tables and graphs. A user-mthematics approach, by contrast, would havestudents list all 'things'. like goals, events, directions or ways of travelling that areessential in making their own holiday plans. Afterwards they are asked to orderthese 'tlungs', to produce their own preferced way of travelling. This may lead toa comparison of the costs of renting and driving the two different cars, hut it couldalso lead to the conclusion that some of the goals have to be changed in orderto keep costs as low as possible, or that cost is of no importance at all. Inuser-mathematics, students are not judged on how they conform with standardrules of mathematical argument, but on the legitimacy of their own producedreasoning.

A recent projectiimed at high-school students in the Netherlands, hasdeveloped integrated mathematics activities which tit well with the ideas of user-mathematics. Students are allowed to choose between different kinds of tasks

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Alarjolijn Witte

(Obdeijn, Dalhoeven and Johannink, 1990), and the acceptance of work aboutcartoons, a light-game, and phases of the moon shows a break with currentstraight-jackets.

Conclusion

In this chapter I contend that gender differences in interest in mathematics maybe difficult to change if we confine our strategies to changing the individuals ortheir environment. In many studies it.has been demonstrated that approaches likethese nnght change the landscape, but the differences remain. The same cautionpertains to situations where one allows for sonic variability in the autonomy ofthose involved, but assumes that gender differences are dependent only on the waylearning materials are delivered, for example, in a friendly or an authoritariansetting.

In contrast, I have proposed a third strategy based on the notion that genderdifferences in mathematics education are entirely system-induced. I have definedthis strategy by describing one way of changing the 'language' of the educational

system and hence the dominant conception of mathematics as currently presented

in most schools. I have suggested that we should explore basing mathematicsteaching more on a constructivist form of mathematics. I have argued that this will

increase the variety of students involved in mathematics. I did not argue, however,

that this strategy is sure to hit the nail on the head.I want to conclude by mentioning sonic new opportunities. There is a

widespread interest in constructivist approaches to educationind this language

already has a tine and extensive implementation in constructivist mathematicseducation (see. for instance. Ernest. 1991; von Glasersfeld, 1991). There are manyopportunities for applying the available instruments but again. I have to stress,as I have done betbre, that it is easy to IA when one does not go the whole way,and also apply constructivist mathematics.

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CON! R.Fy, J. (1990) 'What constructivism implies for teaching', in DAvis, R.B., MArC.A. and Noin>INGS, N. Constructivist Views on the li.whing and Learning oplatheinatics,Journal for Research in Mathematics Education, Monograph no. 4, pp. 107-22.

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SI II KMAN. J. (1981) .md boys' enrolliimus m theoretical math courses: Alongitudinal study', Mpliolo,t9' (I. II ;,man Quailed), S 5 pp. 681-9.

24.3

Marjoliftt Witte

SHERMAN, J. and FENNEMA, E. (1977) 'Thc study of mathematics among high school girlsand boys: Related flictors', American Educational Research Journal, 14,2, pp. 159-68.

Su 1.11Elil.AND, R. and HOYLES, C. (1988) 'Gender perspectives on Logo programming inthe mathematics curriculum', in Hoyt Es, C. (Ed) Girls and Computers: General Issues

and Case Studies of Logo in the Mathematics Classroom, London, Bedford Way Papers, 34,

pp. 40-63.TURKI E, S. (1984) The Second Self: Computers and the Human Spirit, New York, Simon and

Schuster.Vos, E. (1988) 'Wiskunde leren in groepsverband'. Didaktiq; mei, pp. 10-12.WAI KERDINE, V. (1989) Counting Girls Out, London, Virago.Wi ii E, M. (1992) 'Euclidean constraints in mathematics education', Proceedings qf the

Sixteenth PME Cot!ferent-e, Durham, NH, International Group for the Psychology ofMathematics Education, pp. 114-21.

Wi r E. NI. (1994) Meijes meegerekend. De construaie tun iviskundige beganfdheid (Girls count.

The construction of mathematical competence), Amsterdam. Thesis Publishers.

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1. t )

Epilogue

We use the chapter of Nancy Shelley as our epilogue for it leads us to reflect howmathematics might look in the final phase of reform. Beginning from theassumption that we all carry with us a measure of reality which derives from theculture of western mathematics, Nancy Shelley stimulates us to imagine amathematics which is not dominated by the experience of white men. Shechallenges the power of form and authority by adopting a non-traditional writingstyle. She calls to her aid T.S. Eliot's Four Quartets in seeking to summon thoserealities she wants to acknowledge. She asks fundamental questions about the statusof knowledge and how it is culturally influenced, the epistemological status ofissues in mathematics and mathematics teaching such as the role and function ofabsolutes, prediction, otherness, authority, objectivity. Describing her earlyexperience of male domination in mathematics education, she makes a strong pleatbr challenging the relation of mathematics to war and calls for the necessity tofight for peace. She closes with a vision of a different mathematics, which growsout of a culture which has no pretension to dominance, which affirms humanness,and which relies on a culture of relationship, care and communion. According toShelley the process by which it will grow is known, but the shape it will takeremains to be deternUned.

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Chapter 27

Mathematics: Beyond Good and Evil?

Nancy Shelley

Each of us carries with us, at any one time, a number of realities, and it is throughthe interlocking of those realities that we conie to understand the world aroundus. Sometimes we consciously limit what we allow to impinge upon us, in orderbetter to handle a particular reality pressing in its urgency. 'Comprehension',however, as Hannah Arendt' indicates, 'does not mean denying the outrageous,deducing the unprecedented from precedents, or explaining phenomena by suchanalogies and generalities that the impact of reality and the shock of experienceare no longer felt'. It will come only through acknowledging other realities, boththose of others and our own.

All of us carry with us a measure of reality which derives from the culture ofwestern Mathematics. Since that culture permeates world views, values, ways ofthinking, and its influence on attitudes is widespread, we need to face andunderstand realities which draw their nourishment from its bones.

I have used T.S. Eliot's Four Quartets to assist me in seeking to summon thoserealities we here would want to acknowledge. Just as he continually evokes image-cand thoughts which require us to ponder and think beyond the words used, I inviteyou to ponder similarly, allowing the ripples, which move out from the pebbles Idrop into the pool of your listening, to stir the memory of your own experience.2

In this century, we have experienced the largest and most profounddevelopment in science known in recorded history, and the consequent techno-logical explosion penetrates almost the entire globe. We are aware of an evolutionin mathematical thinking of tremendous proportions, although less able either todetect the full extent of its present effect, or perceive its future ramifications.Changes wrought by moving from the absoluteness of space and the absolutenessof time to the concept of a spacetime continuum have yet to impact upon ourconsciousness. We know of differences in mathematics education which affectmore than the skills we employ. These dig deep into our individual philosophiesand impinge upon our understanding of mathematics itself.

Time present atui thne pastAre both prrhaps present ht thne.fithin,And tinwlitture contahwd in time past.'

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Nancy Shelley

In this century, too, the world is experiencing the largest displacement ofpeople from their homes in recorded history; the highest number of people facing

hunger, malnutrition and death from starvation; levels of violence withinoa wa_cacommunities, between nations. within nations, in homes and glol 11 ..y 1 1 are

unprecedented; rampageous destruction, degradation and pollution of the envi-ronment; species elimination on a scale not previously known. All of whichco-exists with the greatest degree of universalist power ever exercised, yet whichis accompanied by a wholesale disaffection with known political processes.

Time past and time film'Allow but a litth. con5ciou5ne5s:4

In the last decades of this century, it is possible to observe a burgeoning: in there-awakening of cultural practice spawning new tbrms derived from old traditions;in the re-learning of human dependence upon the resources of the planet and

interdependence with other life forms; in the re-appraisal of new shibboleths andold verities; in the re-discovery of the richness of understanding 'otherness' anda reahation of the power of respectfulness; in a re-assurance that a polity willoperate effectively only when it is centred round the people who form it.

At the still point of du. turniv world.Neitlwr.fiesh nor.fleshless;

Neitherfrom nor towards; at the still point,

there the dance is,

But neither arrest nor movement. . . .

Except_lOr the point, the still point

There would be no dance, and then. isonly the datwe.5

In the List decades of this century we are witnessing the re-emergence of ethnicrivalries stemming from submerged identities: some being retrieved from centuriesago, some recasting a continual struggle, some being revived after half a century'sincorporation. We eau see the outbreaks of antagonism toward immigrant peoplesreinvoking xenophobic fearind We can see the regathering of opposition torepressive governnwnts. Some of these manifestations exhibit new ideologieswrapped in old myths, some coalesce around an aggressive nationalism and nunyhave become open markets for the arins trade. We can see the repercussions ofeconomic ideologies bringing trauma through disintegration or from gainingascendency. We c,m witness, too, the demands of indigenous people tbr recognitionof their claims to land, their insistence that their dignity and their cultures beacknowledgedmd, we c.ui Cee some fragile attempts of the descendants of theinvaders to appreciate their wisdom.

2-18

II 'outs, after speech, reach

Into the sileno.. ()nly by theiOrm, the pattern,Can words or music reach

'1 'lu. stillness, as a Chilies(' jar

loves perpetually in its stillness."

tr.


Developments of such magnitude, occurring concurrently, demand that anyattempt at understanding, and locating our place in it all, must hold the reality ofthese developments in creative tension. Alongside ways of thinking lie ways ofdoing; alongside operating globally lies living locally; alongside dreams ofdomination lie nightmares of oppression; in, and through, and round about theexercise of power lies massive insecurity. And holding the reality of theserudiments in creative tension can be done only through telling a story.

have been charged with this responsibility, and while any good story willhave many stories within it, the prevailing one must needs be mine, carrying, asit does, the interlocking of my realities. I shall, too, etch in briefly some wordpictures. The reality I derive from western Mathematics stems from teaching it formany years, being involved in teacher education and writing a thesis which wasdealing with how different teaching methods affect the ways in which studentscome to acquire mathematical concepts. Other realities which I carry aregrounded in the power of life-enhancing practices and adhere to my work,particularly over the past decade, in working, writing and taking action for peace.It will be in my story that I hope to encapsulate the things that need to be saidunder my title 'Mathematics: Beyond Good and Evil?'

I asked the men, 'What are you carryingwrapped in that hammock, brothers?' Andthey answered, 'We carry a dead body, brother.'So I asked ... Was he killed or did he die anatural death?"That is difficult to answer,brother. It seems more to have beena murder.' 'How was the man killed?With a knife or a bullet ... ?' 'It wasneither a knife nor a bullet; it was amuch more perfect crime. One that leavesno sign."Then how did they kill this man?'I asked and they calmly answered:'This man was killed by hunger, brother':

Ability

Throughout the early 1970s, in a considerable number of schools I visited, I

observed that not only large numbers of children were failing in Mathematics, butthat they hated it. Teachers were dedicated, burdened and very puzzled. I saw, too,that many teachers relied upon ascertaining whether a child had ability, or not, todecide how that child would be taught.

I loved the subject and greatly enjoyed teaching it. I had always seen that itwas my responsibility to find a way to engage a student in the thinking, and to dothat I searched for the words, or the material, which brought forth his or her ownexperience. I could not believe it was possible to be definitive about whether a

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child was 'able' or 'not able'. And, for me, there is a qualitative difference betweenwhat we can know about objects and what we can know about another humanbeing. German, for example, has the two verbs, wissen and kennen, which indicatethis difference, whereas normal English usage has but one word, although thevestiges of distinction can be found in the language. I am indebted to Werner Pelzfor the following:

Vissen is both 'to know' and 'knowledge'. I know this, that and the other,know how to do this, manage that ich weiss. . . . I may and must wissenhow to run a certain machine. 1 Vism.n is certain or gewiss, yes, or no,binary ... Iffissen gives manipulative power ... It :s a matter of skill, ...precision.

Kennen recognizes the other in his or her or its otherness, ... [what] I canbegin to known, has the character of ... a Lonfession of faith, of anAnerkenntnis or acknowledgment.... Kernwsi implies ... sympathy,empathy, patience. Openness, deference. It is close to love .. .8 [Emphasisin original]

To 'know' a person is clearly distinguished from knowing an object. Themerging of the two conceptual images permits the power and manipulation ofhumans, by other humans.

It is of interest here to consider the sensitivity with which AustralianAborigines approach coming to know about other human beings.

Aboriginal social life is very public and personal privacy is ensured ...through verbal privacy, particularly through an indirect style in muchverbal interaction. ... [I]n situations where Aboriginal people want tofind out Nvha t they consider to be significant or :ertain information, theydo not use direct questions. Inkmation is sor Olt as part (!f a two-wayexchanqe. Silence and waiting till people are ready, . . . are also central toAborigi tal ways of seekiv any substantial information." [Emphasis in original]

Modern . feeling[s] of isolation and powerlessness [are] increased ... by thecharacter which all ... human relationships have assumed. The concrete

relationship of one individual to another has lost its direct and humancharacter and assumed a spirit of manipulation and instrumentality"'

Absolutes

When Newton tbrmulared that.

Absolute space, in its own nature. without regard to anything external,remains always similar and immovable,

and that

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lathematics: Beyond Good and Elil?

Absolute true and mathematical time, of itself, and by its own nature.flows uniformly, without regard to anything external.'

he laid the foundation for the acceptance of the primacy and uniqueness ofsequential time and for the replacement of the Aristotelian conception of spacewith one which henceforth became identified with the real space of the world.To acknowledge the absolute nature of space and time is to set them apartfrom humans just as God had been set apart. To receive their absolutenessas true is to enable constructions built upon them to be separate from humanexperience.

As Polanyi put it:

the mechanical properties of things alone were primary ... their otherproperties were derivative or secondary ... This too was a theoretical andobjective view, in the sense of replacing the evidence of our senses by aformal spacetime map that predicted the motions of material particleswhich were supposed to underlie all external experience.'

This 'formal spacetime map' yielding its predictions of 'the motions ofmaterial particles' did not exempt humans in its majestic sweep 'to underlie allexternal experience'. As Popper noted:

Most openminded menind especially most scientists thought that in theend it would explain everything. including ... even living organisms.'

So it came about that humans, too, could be observed and measured; their actionspredicted; they were likened to machines and the power of mechanism andphysical determinism knew no bounds.

The knou,ledge imposes a patteni, and

For the pattern is lull, ill every moment.-lnd every moment is a new tind shocking

I;duation of 'ill WC haiw been.14

The teaching/learning process, particularly that relating to Mathematics, hasalso been affected by mechanism. When that process relies on seeing humans asbeing capable of being observed and measured by other humans through designof tests, through psychological practices, it is employing a mechanistic view ofhuman beings. and then, more than the learning of Mathematics is affected.Questions of self-image are involved, and, of confidence which seeps through toother areas of students lives.

It I violate ... basic human dignity. I am being violent.'

From the noted Australian historian, Henry Reynolds, we learn that;

'3 '.S.

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Nancy Shelley

Aboriginal self-confidence was notbased solely on the mastery ofparticular skills but on the spiritualrelationship with the land, the sense ofbelonging and responsibility ...le'

I'rediction

Another Aement resides within our urge to 'know in advance'. The basis ofprediction lies with our notion of causality and present ideas of causc and effectrely upon the acceptance of sequential time given primacy by Newton.Aristotelian ideas of cause were teleological wherein 'the "nature- of anything isnot its first but final condition' toward which it progressed, joward perfection.17Instead of cause 'drawing you towards', it switched to 'driving you from behind'

the thing that drove then became more 'r!ally' real than you."'

Here the impossible 1h:ionOf spheres (f existence is actual,Here the past and futureAre conqiwred, and reconciled,Where action were otherwise movementUf that which is Only mopedAnd has in it no source of movementP)

To come to know in ways that allow more and more to be foreseen, bestowsa sense of power. It brings, as well. a sense of fear. As power from predictionincreases, and when that power of intellect is transmuted through dominance topower political. 'tis then it seeks control. Only this year, in response to a lawenacted in Washington State, the US Society of Psychologists found it necessaryto make the public statement: 'You camiot predict future criminality.' Meanwhilethere is little registering the accompanying insecurity which pervades the need toknow events ahead of experiencing them.

Otherness

Throughout the period when I was writing the thesis, I was engaged in threeparticular programmes of mathematics teaching: one involved senior high-schoolstudents who had 'dropped out' of normal schooling, one involved women whoseprevious experience was that of having 'failed', and the third involved a pilotscheme with Aboriginal children who had been labelled 'unable'. Together theseexperiences in( reased my understanding of what WA', embedded it) the subjectcalled Mathematics and what was built into the concept of failure in the subject.

In each of these three situations, I placed .1 priority upon establishing anatmosphere of trust, recogniimg the other in his or In r otherness. By an 'opening

7 52

,


up' and a 'dras ig in'2" to an active experience of relation one to another, and ofrelation of each to the 'matter' in mathematics, it was possible to develop a newunderstanding of it. As the criteria for 'correctness' were re-established withineach of the participants, that is, as they learned to take responsibility for decision-making, a process of learning took place, and mathematics became a meaningfulexercise. The association with which they came to know affected what it was thcylearnt. Mathematics began to reveal some of its richness and became enjoyable asit offered new ways of thinking.

Authority

In my work in teacher education, I had noted that the diversity which the studentsdisplayed presented a problem for the school and for the teachers, and indeed, ithad become a pedagogical nuisance. At the same time. I came to realize that astrong factor which affected teachers was the authority the subject held for them,while a coninion perception of Mathematics in society MS its rock-likeirrefutability.

For me, diversity is something to celebrate as it brings into any experience theunexpected, which c:instantly reminds me of the ever-renewing capacity of livingbeings. Besides, when diversity appear, a problem, uniformity is lurking in thewings!

Desio-ation oldie world of sense,Erafflation of the world offantasy,Inoperancy of the world of spirit;

. . . while the world movesIii appetesicy, on its metalled uuys()f time past and time future.21

The world of spirit was fir fi-om inoperative throughout the eighteenth andmost of the nineteenth centuries, nor had the world of fantasy been evacuated. asmathematical insights came and ideas were born, as one writer described it, 'in averitable orgy of intuitive guesswork'. The 'ecstasy of progress' opened up 'amathematical world of immense riches'.22

It \SW, not until the end of the niieteenth century that there was a return tothe classical Greek ideal of precision and rigorou, proof: to the employme»t of'logically precise reasoning starting from clear definitions and non-contradictory-evident- axioms'.2 With die arduous work of the Logicists, working on theseveral 'pieces', we had a shaping of those 'immense riches', and all otheis thatfollowed them, in an attempt to thshion a whole a hierarchical whole which,it wa., hoped, would exhibit the sanw characteristics: consistency throughout,using argument based on logic, and, upon completion, the acknowledgment ofvalidity. The result has been a very substantial, indeed liirmidable, cdiii . Yet, thesum of innumerable distinct fragments of validity does not constitute a verdict oftruth.

7.5.3

Nancy Shelley

If the logic employed has, concurrently become the basis for public argumentand accepted rationality as it has it also allows the whole to assume anauthority which is daunting and a visage of irrefutability which exudes greatpower. It was the authority of Aristotle which played a large part in the restrictingof human thought for centuries. 'Tis this that most people encounter whenintroduced to western Mathematics. 'Tis this which has earned its description asan objective body of knowledge. 'Tis this which further disconnects it from humanexperience.

Objectivity

People now . .. tend to trust their own experience, their eyesand ears, less than a system which by virtue of its

consistency is more rational than reality.24

In the opinion of many mathematicians thi:: objectivity constitutes a strength; formost learners it is an impediment; for many teachers it is inhibiting and for muchtea,thing/learning it becomes an imposition. For the subject itself, it is damaging

it separates it from its roots, falsifies its history; and has dramatically influencedthe direction its development has taken. Meanwhile, it ignores how mathematicalideas originate and limits mathematical thinking by channelling it in selectedpaths.

Also, in its wake it brings a number of characteristics. It puts ;! high value onconsistency and logic; validity is prized. It is hierarchical. Through systematizinghow Mathematics is to be viewed, the basis for decision-making has been takenout of the hands of teachers, of students, and, indeed, of all humans who come incontact with it; 'correctness' hereafter resides within the system it becomes theauthority by which everything is decided.

Any statement made out of objective reality comes across as a command; itdictates; certain conclusions are 'necessary'. The system itself is now one of staticpermanence; its validity has become truth, andis Hannah Arendt observed:

the kind of truth derived from the paradigm of truth dependent upon thelogical argument which informs western thought, is essentially coercive

and, of its nature silences opinion and tries to impose undoi

Through its logic this Mathematics has become coercive and imposesunitbrmity, for

... the truth of a systein . . jcomesi to be a threat to freedom ... because itallows no room for hunimi diversity.2"

Acceptance of this visage requires an imposition also, for it prescribes whatshall be learnt, and, that does not derive in any way from one's experience; itrequires compliance.

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Mathematks: Beyond Good and Evil?

The more one conforms to anonymous authorities,the more one adopts a self which is not one's own,

then, the more powerless one feels andthe more one is forced to conform.'

If we remove humans and the earth from relationship, we turn both into objectsto be acted upon; this is violence. This Mathematics holds over the heads ofprospective learners impending decisions concerning their ability, particularlyrelating to how logical they are. It dominates. It seeks obedience. It seeks control.The subject is then seen to be more important than a student's engagement in it.

This story of a Korean woman expresses it clearly:

The major focus in the curriculum was learningmathematics and English and other foreignlanguages ... Good grades automatically led us to goodcolleges. We frantically memorized theforeign vocabulary and mathematical formulas likerobots so we could get good grades. It was a long andlonely adolescence.28

Male Domain

I, too, was beginning to feel lonely as I endeavoured to tease out and put flesh onthe bones of my thinking, for I could not find mathematics people who knew whatI was saying. Then, in 1976. I came to my first ICME, in Karlsruhe, and, thinkingto test the waters, I offered a 'Short Communication'. Two things stand out forme from that Congress.

At a special session which was hastily arranged, along with others, I wasinvited to speak. There ensued a dialogue with some of the participants which wasboth lively and affirming. That dialogue was most important for me because,instead of feeling isolated. I learned there were other people across the globe whowere exploring similar paths.

The second thing that stands out for ine from that Congress was theextraordinary feeling I had that it was totally male-dominated, despite the presence ofa large number of women participants. I experienced very strong feelinp of beingpushed down and being held there, without any opportunity to comment on whatwas happening. I checked out my impressions with some Australian colleaguesboth female and male and found that they, too, felt it. So we decided to dosomething about it and called a mee.ing of women 'to talk about ICME'.

About seventy-five people came, both women and men, and out of thatmeeting IOWME, the International Organization of Women and MathematicsEducation, was born. IOWME has affected the format of each ICME since, helpedto bring the question of women and Mathematics into the arenamd now hasbranches in more than forty countries.

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Nancy Shelley

However, the path by which we moved from first calling that meeting to theestablishment of IOWME and the acceptance of its tirst resolution, which wesought to present to the final plenary, was not a smooth one. Sonic women were .actively discouraged from attending the meeting by male colleagues from theirown country. Attempts were made to break up our meeting. We had to dealand I choose my words carefully with anger. alarm, obstruction and pilblicderision.

Of course, I WAS aware that Mathematics was considered a male domain. I hadexperienced non-acceptance at times, and knew that most girls did not doMathematics. However, here was a direct experience of being oppressed by theauthority of a large number of men who, very visibk exercised control overproceedings, and apparently over what was acceptable as to how mathematicseducation should be developed. I had not before experienced the power ofownership which men, eminent in their field, sought to exercise.

From time to time. particular groups of people with common interests cometogether with the specific purpose of furthering the development of anunderstanding of the world along certain particular paths. Western Mathematicsis the product of the work of such a group, and it constitutes a very elaborate modelof reality built up over several centuries.

As Ursula Franklin noted:

Models and analogies are always needed for communication, and in orderto be useful tools for discussion, models and metaphors need to be basedon shared and commonly understood experiences.'"

Western Mathematics has moved away from 'commonly understood experi-ences'. Its beginnings lie in the common experience and culture of European men.and constitute a model which they found enabled them to interpret the world.Much has been built upon those beginnings; it has, indeed, become an extremelyelaborate construction. It remains, howeveri modelilbeit a very powerful one.

If a map is mistaken for the territory itself unreality becomes its monarch. Ifthis model from western Mathematics is confused with reality itself, validityis translated into truth. what is one model only becomes the only reality andA static one and the intellectual power which the model bestows gives way torigidity requiring defence: to convince assumes importance and justification.compelling.'"

As Barbara McClintock recalled:

I couldn't tell other people at the time because it was against the 'scientificmethod'. ... we just hadn't touched on this kind of knowledge ... [andit is] very different from the knowledge We call the only way."'

rr'a, 0, go, mtid the bird: lnunaii kimlCannot bear wry much reality'

Now any model of reality has power for those who construct it and a modelshared becomes the basis of a culture, centred round the reality which the model

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Mathematics: Beyodd Good and Evil?

affords, and bestowing its power. The construction of western Mathematics arisingout of the culture of European men, and, having, as it does, their 'historicallydetermined values built into it', as Mike Cooley expressed it, it must needs be amale domain." It is not surprising, therefore, in the presence of the heirs andarchitects of that model, to experience a sense of ownership.

I said to my soul, be still, and wait without hopeFor hope would be hope for the wrong thing; wait

without love

For love would be love of the wrong thing; there is yetPith

But the.taith and the lotv and the hope are all in thewaithig.

without thought,Pr you are not readyfor thought:

So the darkness shall be the light, and the stillnessthe dancing.'

Peace and War

What a bewildering world ...They feed people with words,the pigs with choice potatoes.

Each daythout 300 km' of tropical forests are cut down,and, each day, around 160 km' ofland turns to desert.

114. must be still and still moving

Into another intensityFor a fttrther union, a deeper cotnnumion

Throttgh the dark cold and the empty desolation,

In my end is my beginning.'"

The thesis was duly completed and it led me on to articulate more clearly whatI see as central to the teaching/learning process, to develop new methods ofteacher education and to run workshops for practising teachers.

Then, in the early 80s, I mad, the decision to give up paid employment inorder to work full-time for peace. In doing this I was giving precedence to a realitywhich had become more and more important to me. My commitment to peacedid not arise out of fear of the possibility of war, but is grounded in myconvincement of the power of life-enhancing practices.

In most western minds throughout the 80s there was a preoccupation with thepossibility of annihilation which came out of the heightened confrontationbetween the western powers and the Soviet bloc, made feasible by the enormityof opposing militar) juggernauts.

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Nancy Shelley

The number of people dying [in the world]as a result of malnutrition is equivalent todropping a Hiroshima bomb every three days.[Emphasis in original]'

The engineering of the world's annihilation goes hand in handwith the engineering of the world's consent.'s

When massive numbers of people withdrew their consent, the superpowersbacked away from the brink.

Towards the end of the 80s came a rapprochement between the superpowers,and as glasnost affected the thinking in both east and west, the visage of 'the enemy'dissolved. This brought much rejoicing which swelled in crescendo with thecrumbling of the Berlin Wall. There was, however, a murmur of insecurity

People change, and smile: btu the agony abides.'

When the crumbling became the symbol of a vaster internal fragmentationthat sought its perestroika, the murmur of unease became persistent. There was littleunderstanding that strategists, the military and the industrial complexes, whichdepend upon governments' subsidization, were working from different premises.

Militarism has proven to have a particularly powerfulideological pull.'"

ithere is there an end of it, the soundless mailinq,The silent wither* qf autumnflowersDropping their petals and remaining motionless;IVhere is there an end to the drifting wreckage-4 I

During the first half of 1990, the Pentagon devised its new military strategyof mid-intensity conflict to replace that of high-intensity previously directedtoward the Soviet Union.

There is no end, but addition-42

In the first half of 1991 we witnessed the Gulf War, which marshalled thecombined forces of many countries. While the Iraqi people faced the reality ofbeing bombarded, on TV screens throughout the world, viewers were bombardedwith a virtual-reality display of the 'theatre' of war. Weapons took on humanattributes: 'smart' weapons had 'eyes', compwers 'brains' which 'made decisions',and it was military targets and missiles which were 'killed'. Human beings acquiredthe characteristics of objects: if they were military personnel they 'absorbedmunitions'; if civilian, they were not hurt, maimed or killed but recorded as partof 'collateral damage'.4'

258


There is no end of it, the voiceless wailing,

No end to the witheriv of withered flowers,the momment of pain that is painless and

motionless,

M the drift of the sea and the drifting wreckage

The repercussions of that war will continue to reverberate for many years andin many ways. Few reactions will surpass the effect of the dehumanizing that tookplace: through people's senses, through being an audience assisted by thatextraordinary switch of language through the total exclusion of reference tomaiming or killing of human beings and through the absence of outrage. Thedesensitization will have found its way into the crevices of people's minds.

the excessive power of a few leads to the dehumanization of all.45

. . we come to discot,er that the moments of agony

. . are . . . permanentWith such permanence as time has.'

At a NATO meeting in November 1991, the murmur was openly articulatedwhen George Bush announced that since the loss of 'an enemy' we now have'insecurity' and 'instability'. From the perspective of power, it appears that securityand stability derive from having 'an enemy'.

Do not let me hearOf the wisdom of old men, but rather of their Ply,Theirfrar el:fear-4'

The values embraced in militarist ideology have takenon their own life and their own laws.'

It seems, as one becomes Met;That the past has another pattern, atid ceases to

be a mere sequence

Or even development: 49

From my past I began to see another pattern. Through my research intomatters relating to war and peace, I learned that by fiir the greatest number ofmathematicians are engaged in military research and development. In con-sequence, the single gre itest area of mathematics work, today, is problem-solvingfor the military; the foundations of military science are underpinned by westernMathematics.

Mathematics, of course, can be put to innumerable uses; some of those usesare disowned by mathematicians, sonic are manifestly invalid. Here, however, thereare different matters to consider. First, since the military attracts, and funds, themajority, and those designated the brightest, of today's mathematicians, we need

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Nancy Shelley

to realize that the major development of mathematics is now channelled in veryselected ways.

Another matter germane to us here is why so many mathematicians fit readilyinto a particular type of research? What makes it conducive for one trained inmathematical thinking to be at ease within that cast found in a military culture?

We live in a world where western culture is dominant; in a world wheredecisions made are described as logical and rational; where power is exercisedthrough hierarchies; where authority has clout; where dominance is the norm;where power is exercised by the few over the many; where order is imposed by'knowledgeable' people; where power is enforced by military might.

Western Mathematics plays a dominant role in western culture. People withmathematics qualifications are highly respected; they enjoy status; their work isvalued as a necessary and integral part of society. The logic which permeateswestern Mathematics, having been the basis of accepted rationality in westernthinking for centuries, is now pervading the globe. Dealing with problems, it iscommon practice to redefine them in such a way as to facilitate their mathematiza-tion.5" The culture of western Mathematics derives its strength from thedominance of its model of reality. This culture is now a dominant culture in theworld, and, as with any dominant culture, it lays claim to universality, to actualityand to uniqueness. Western Mathematics is value-laden.

Every relationship of domination ... is by definition violent, whether or notthe violence is expressed by drastic means. In such a relationship,

dominator and dominated alike are reduced to things the formerdehumanized by an excess of power, the latter by lack of

We live, too, in a world where the power of the military is great: wheremilitary 'solutions' are seen to be most effective, and the carrying out of ordersmost efficient; where training develops a particular reality which becomesmandatory; where problems are 'refined' to eliminate ambiguity; where theprecision of military decision-making is admired anucl the complexities of humanissues; where training of personnel involves 'killing the other' in them; where themilitary is a major contributor to land degradation; where the military is a primarysource of environmental poisoning; where it is recognized that an efficient killeris one who no longer contemplates the humanness of the enemy.

Here between the hither and the further shore

While time is withdrawn, wnsider theihtureAnd the past with an equal mind.52

Concurrently, in our world, increasingly, we can find more and more people:living frugally in order to be more responsible; regaining dignity through creativeart; designing and constructing energy-efficient buildings; establishing commu-nity; developing relationship with the land; devising practical schemes ofassistance; restoring ecosystems; renewing relationships within small groups;repairing enviromnental damage; networking with others locally, nationally andinter nationally.

260

Alathematics: Beyond Good and Evil?

Time past and time fitureIVItat might have been and what has beenPoint to one end, which is always present.'

There is, too, the possibility of a different mathematics which grows out of aculture which has no pretensions to dominance; which can yet engage a learnercreatively; which affirms humanness; and which attends human needs and willembrace earthy things. A culture of relationship, of care, of communion. Theprocess by which if will grow is known, the shape it will take remains to be made.

To become renewed, transfigured, in another pattern.'

Today, too, we live in a culture of violence. A culture: where to dominate isto be strong; where power is maintained by violence; where to conquer is to standtall; where some lives are counted more valuable than others; where decisionsmade to expend resources upon weapons are called rational while millions starve;where torture is frequently applied; where there is violence against the 'other',against the poor, the hungry, women, the young, racial violence; where humandignity is constantly violated; where competition is promoted and to win isessential, the loser despised; where conditions are not negotiated, but imposed;and, where the violence of repression, of exploitation, of unemployment, ofoppression, and destruction of the earth are widespread.

I 'hat we call the beginnhig is often tlw endArid to make an end is to make a beginniq.Tlw end is where we start from.'

Competing realities of such magnitude, occurring concurrently, require thatwe constantly hold them in creative tension, if we want to attempt undersonding.Each of us will take our place as we see fit; different realities will have different pullsupon us.

To postpone or evade the decision is still to decide.To hide the matter is to decide. lb compromise is todecide. There is no escape from this decision ..

Is Mathematics beyond good and evil?

In order to arrive a( what you do not knowYou must go by a way which is the way of-ignorance.

In order to possess what you do not possessl'ou must go by the way of-dispossession.

In order to arrive at what you are notmust go through the way in which you are not.

And what you do not know is the only thing you knowAnd what you own is what you do not ownAnd where you are is where you are not.'

;.1:1261

Nancy Shelley

Notes

1 AREND I; H. (1973) The Origins of Totalitarianism, New York, Harcourt BraceJovanovich, page viii.

2 'Mathematics: Beyond Good and Evil?' is written to appeal to more humancharacteristics than logic. One of those is hearing. In order, therefore, to hear itsrhythm, the author reconunends that the chapter be read aloud. The author wasinvited by the organizers of ICME-7 to present this paper as one of the Lectures atthe ICME-7 Congress, in Québec City, in August 1992. It was well received. TheCommittee responsible for publication of those lectures declined to publish it. So theeditors of this book, which is based on papers presented in the IOWME sessions atICME-7, decided to include it.

3 EuoT, T.S. (Mcmlviii) Four Quartets, London, Faber and Faber, page 7.4 Ibid, page 10.5 Ibid. page 9.6 Ibid, page 12.7 BENNETT', J. with GEORGE, S. (1987) The Hunger Machine: The Politics qr. Food, New

York, Polity Press, page 21.8 PEI z, W. (1974) The Scope qf Understanding in Sociology: lbwards a More Radical

Orientation in the Social and Humanistic Sciences, London, Routledge and Kegan Paul,pages 80-81.

9 EADES, D. (1992) Aboriginal English and the Lau; Continuing Legal EducationDepartment of the Queensland Law Society Incorporated, page 28.

10 FRomm, E. (1960) Fear of Freedom, London, Routledge and Kegan Paul, page 102.11 NEWTON, I. (1970) 'Philosophiae Naturalis Principia Mathematica', in BAS, C. VAN

FRAASEN, An Introduction to the Philosophy of Time and Space, New York, RandomHouse, page 23.

12 POLANYI, M. (1973) Personal Knowledge: Tbwards a Post-Crkkal Philosophy, London,Routledge and Kegan Paul, page 8.

13 PoppER, K. (1972) Objective Knowledge: An Evolutionary Approach, Oxford, ClarendonPress, .page 211.

14 Euar, op cit, page 18.15 ZARU, J. (1986) 'May God's peace, mercy, and blessings be with you', in ECK, D.L. and

J. (Eds) Speaking qf Faith: Cross-Cultural Perspectives on Women, Religion andChange, London, The Women's Press, page 54.

16 REY NOI DS, H. (1982) The Other Side of the Frontier: Aboriginal Resistance to the EuropeanInvasion of Australia, Ringwood, Penguin, pages 58-59.

17 AR lsi MIT (1964) The Politks, translated by TA. Sinclair, Harmondsworth, Penguin,in the Introduction, page 28.

18 Ht-1 I Elk, E. (1961) The Disinherited Mind, Harmondsworth, Penguin, page 186.19 Ei KY!, op cit, page 33.20 Buitiat, M. (1971) Between Man and Man, London, Collins, page 118.21 Et lot, op dt, page 11.22 COUILAN m, R. and ROBBINS, H. (1948) What is Mathematics?: An Elementary Approach

to Idras and Methods, London, Oxford University Press, page xvi.23 (bid, page xvi.24 CANOVAN, M. (1974) The Political Thought of Hannah Arendt, London, Methuen, page

21.

262

. t.)


25 Ibid, page 114.26 Ibid, page 114.27 FROMM, E. quoted in FRE1RE, P. (1973) Education For Critical Cnnsdousness, New York,

Seabury Press, pages 6-7.28 CI RiNG, H.K. (1990) Struggle to be the Sun Again, Orbis Books, page 2.29 FRANK! IN, U. (1990) The Real World of 'Thchnology, Massey Lecture Series, Montréal,

CBC Enterprises, page 26.30 Adapted from MURA. R. (1991) Searching For Subjectivity in the World of the Sciences:

Feminist Viewpoints, CRIAW/ICREF, page 52.31 McCuNTocK, B. quoted in KE1 LTA, E.F. (1983) A Feeling for the Organism: 1he

Life and IFork of Barbara McClintock, New York, WH. Freeman and Company, page203.

32 Ei loT, op (it, page 8.33 Cool EY, M. (1980) Architect or Bee? The Human ni.chnology Relationship, Sydney.

TransNational Co-operative Limited, page 43.34 Dim, op (it, page 19.35 BENNE1, op cit, page 37.36 Et or, op cit, pages 22-23.37 BTNNLT. op dt, page 12.38 Ot mo, C.P. in the Introduction to Chomsky Noam. (1981) Radical Priorities,

Montréal, Black ROse Books. page 13.39 FA tot, op th, page 29.40 SrAI in, D. and Smu H. R. (1983) 17w Economics of Militarism, London, Pluto Press. page

50.41 Et 101, Op cit, page 27.42 Ibid. page 27.43 Adapted from Col IN, C.E. (1991) 'Decoding Military Newspeak'. in Ms.: 17w

of Women, 1.5, page 88.44 Ei to 1, op cit, page 28.45 Ftu to , op (it, page 10.46 Et or. op cit, page 29.47 Ibid, page 18.48 SrAlri I and Swill, op cit, page 52.49 El tot, op cit, page 28.50 Multn, op cit, page 54.51 FRI IRL, op cii, footnote on page 10.52 Et to 1, op eit, pages 30-31.53 lbid, page 8.54 Ibid, page 41.55 lbid, page 42.56 Zmw, op cit, page 55.57 El tot, op dt, page 20.

References

AR! NI ) I, H. (1973) '17w Origins of 1lnalitarianism, New York, Harcourt Brace jovanovich.

263

Nattcy Shelley

Akis ton a (1964) The Milks, translated by Sinclair, TA., Harmondsworth, Penginn, inthe Introduction.

J. with G1:01U;1%, S. (1987) The Hunger Machine: 71w Politks of Food. New York,Polity Press. .

Bum R, M. (1971) Between Man and Man, London, Collins.CANOVAN, M. (1974) The Politkal Thought of Hannah Arendt, London, Methueq.Ci IUNG, H.K. (1990) Straggle to be the Sun Again, Orbis Books.Coos. C.E. (1991) 'Decoding Military Newspeak', in Ms.: The II 'odd Of li 'men, 1, 5.Cool ry, M. (1980) Architect or Bee? The Human ai.chnology Relationship. Sydney. Trans-

National Co-operative Limited.CoukANI, R. and Roiturs:s, H. (1948) liLit is Mathematics?: An Elementary Approach to

Ideas and Methods, London. Oxtbrd University Press.EAI us. D. (1992) Aboriginal English and the Law Continuing Legal Education Department

of the Queensland Law Society Incorporated.Et R )1, TS. (Mcmlviii) Four Quartets, London. Faber and Faber.FRANKI IN, U. (1990) 71w Real World of "Ii4mology, Massey Lecture Series. Montreal. CBC

Enterprises.Flu nom, E. (1960) Fear of Freedom, London, Routledge and Kegan Paul.FRowina. E. quoted in FRI . P. (1973) Education For Critical Consciousrass, New York,

Seabury Press.Hi ilak, E. (1961) Tlw Disinherih d Mind, Harmondsworth. Penguin.Mc QIN I )(1:, B. quoted in Ki E.F. (1983) A Feeling for the Organism: Tlw Life and

Z)rk ,f Barbara McClintock. New York, W.H. Freeman and Company.MORA, R. (1991) Searching For Subjectivity in the liMd of the Sciences: Feminist I7ewpoints,

CRIAW/ICREF.Ni W I ON, J. (1970) Thilosophiae Naturalis Principia Mathematica'. in VAN FRAAM N. BAN

C., Ali Introduction to the Philosophy of Time and Space, New York, Random House.C.P. in the Introduction to ClionasKv. N. (1981) Radical Priorities. Montreal, Black

Rose Books.1/, W. (1974) "Ille Scope of I'mierstanding in Sociology: Tinvards a More Radical Orientation

in the Social and Humanistic Sciences, London. Routledge and Kegan Paul.Pc )1 AN:VI. M. (1973) Personal Knowledge: Tneards a Post-Critical Philosophy London.

Routledge and Kegan Paul.I); win is. K. (1972) Objatif,e Knowledge: An Evolutionary Approach, Oxford. Clarendon Pres:.Ru INoi Ps, H. (1982) Tlw Other Side of tlu. Frontier: Aboriginal Resistance to the European

hnusion of Australia, Ringwood. Penguin.Snai iii. D. and SM: iii, R. (1983) 77w Economics of Militarism, London, Pluto Press.ZARG, J. (1986) 'May God's peace. mercyind blessings be with you', in ECK, DI. and

Di vim, J. (Eds) Speaking of Faith: Cross-Cult:nal Perspectives on II;inwn, Religion andChange, London, The Women's Press.

264 41) .

...

Notes on Contributors

Joanne Rossi Becker has a doctorate in mathematics education from theUniversity of Maryland and is a professor of mathematics at San Jose StateUniversity. Much of her research has been in the area of gender and mathematics,including classroom-interaction studies and interviews with graduate students inthe mathematical sciences. She has been active in professional groups, serving aschair of Women and Mathematics Education, and of a special interest group of theAmerican Educational Research Association on Research on Women andEducation, and coordinating the IOWME sessions at ICME-5 in Australia.

Paul Brandon has a doctorate in educational psychology and is an assistantprofe,,or of educational evaluation at the University of Hawai'i at Manoa. He hasstudied educational achievement among Asian and Pacific Americans, includinggender differences in mathematics achievement among HawaiTs four majorethnic groups and in the educational attainment of Asian Americans nationwide.

Leone Burton is a professor of education (mathematics and science) at theUniversity of Birmingham. She has been closely involved with mathematicsteacher education in England and has experience teaching students of all ages fromthe very young to adults. Her research interests began with problem-solving butdeveloped into a parallel concern with social-justice issues and the nature ofmathematics. Like many others with a professional interest in mathematics, musicis also a passion. She plays (more or less badly) and enjoys listening particularly tobaroque chamber music and to modern jazz.

Francoise Delon works in the field of mathematical logic. She is 'directrice deRecherche' at the 'Centre National de la Recherche Scientitique'in affiliate ofthe University of Paris. She has been active in the association 'Femmes etMathematiques' since its inception in 1987, and was its president from 1989 to1990,

Sharleen D. Forbes, in addition to being the mother of four t hildren, has had over

1)1+ a 1.).

265

Notes OH COHttibutors

twelve years of secondary-school and university experience in mathematicseducation, with a focus on the teaching of statistics. She has been involved inmathematics research for the last decade, with particular areas of interest being'second-chance education in mathematics, the evaluation of gender and ethnicdifferences in mathematics performance, and the assessment of mathematicslearning.

Lynn Friedman has masters degrees in mathematics and in statistics and obtainedher doctorate in mathematics education from the University of Chicago.Currently, she is an assistant professor in the Department of EducationalPsychology at the University of Minnesota where she specializes in meta-analysisand cognitive gender differences.

Olive Fullerton has a doctorate in education from the University of Toronto andis a professor in the Faculty of Education at York University where she teachesboth undergraduate and graduate students. Her research interests include dysfunc-tional attitudes around mathematics, especially in practising teachers and howthose attitudes impact on young children, how young children learn mathematics,and especially the role of oral language in learning mathematics.

Barbro Grevholm has a masters degree in mathematics, theoretical physics,physics and mathematical statistics. She has many years of experience teaching inthe upper-secondary school system as well as lecturing at the MathenutiesInstitution in Lund. Since 1985 she has been teaching part-time at the Sc hool ofEducation in Malmo.

Saleha Naghmi Habibullah was born in Pakistan and has lived for most of her lifein Lahore. She has masters degrees in statistics from both the University of thePunjab and the University of Toronto. She has taught at Kinnaird College forWomen in Lahore since 1979 where she is currently head of the statisticsdepartment. Her primary area of interest is in the devdopment of statisticaleducation at the undergraduate level in Pakistan.

Mary Harris became interested in problems associated with learning mathematicsafter a number of part-time jobs while bringing up her mentally disabled daughter.After obtaining a degree in education, she worked in mathematics education with'low attainers'. Currently she is a visiting fellow in the Department of Mathe-matics, Statistics and Computing a. the University of London Institute ofEducation.

Pat Hiddleston is an associate professor of math..matics education at the

University of Malawi where she is also head of the Department of Curriculum andTeaching Studies. Her research interests are in gender issues in mathematicseducation and in the comparison of attitudes to mathematics between developingand developed countrics.

2664)


Terry Ann Higa has a professional diploma in elementary education, and mastersdegrees in secondary-education curriculum and instruction, and in educationalpsychology. She is a graduate assistant for the Curriculum Research andDevelopment Group at the College of Education in the University of Hawai'i,Miinoa. She recently completed a study of the validity of an instrument on thecognitive approaches of medical-school students.

Joanna Higgins is a doctoral candidate at Victoria University of Wellington andteaches mathematics education at Wellington College of Education/Te Whanauo Ako Pai ki Te Upoko o Te Ika, New Zealand/Aotearoa. She has taught inelementary schools and is currently working with teachers on developing theirclassroom mathematics programmes.

Betty Johnston is a lecturer in numeracy in the School of Adult and LanguageEducation at the University of Technology in Sydney, Australia. She has taughtmathematics to students of all ages, from 5 to 85 years of age, in a variety ofcountries New Zealand, Canada, England, Ghana, Papua New Guinea andAustralia. She has also spent many ars teaching teachers of those students Herparticular interest is in what might be called critical numeracy.

Cathie Jordan holds a doctorate in anthropology and is a researcher andprogranune developer with the Early Education Division of KamehamehaSchools/Bishop Estate in Honolulu. Her current research is on factors affectingfamily utilization of preschool educational prof, .4mmes and on the incorporationof Hawaiian values into educational programmes for native Hawaiian children.

Gincharn Singh Kaeley was born, raised, and educated in India. His teachingcareer began in Kenya in 1963 where he served a, mathematics instructor at theUniversity of Nairobi for several years. Since then, he worked as a mathematicscoordinator in the Department of Extension Studies, University of Papua NewGuinea where he did research on the teaching of mathematics to mature studentswith a focus on gender differences. Currently he is working with the Universityof Chicago School Mathematics Project (secondary component) as director ofevaluations.

Gabriele Kaiser holds a mast,2rs degree as a teacher for mathematics andhumanities and has taught in schools for a few years. She has completed herdoctorate in mathematics education and is currently working at the University ofKassel doing empirical research comparing mathematics education in England andGermany. Co-founder, with Christine Keitel and Cornelia Niederdrenk-Felgner,of the German IOWME section, she now serves as its national coordinator. Jointlywith Pat Rogers she organized the IOWME sessions at 1CME-7. She has a littledaughter and in her spare time she loves to play tennis and do aerobic exercises.

Berindeijeet Kaur is a mathematics-education lecturer in the School of Sciencc

267

Notcs on Contributors

in the National Institute of Education, Nanyang Technological University,Singapore. She has a masters degree from the University of Nottingham and iscurrently working on her doctorate at Monash University, Australia. She is thenational coordinator of IOWME for Singapore and her research interests are in theareas of problems in mathematics and in gender issues.

Hélène Kayler is a professor of mathematics education in the Département deMathematiques of the University of Quebec in Montreal. She has a masters degreein mathematics, specializing in mathematics education. Currently, she is involvedin elementary-teacher education and doing research on the learning of mathe-matic5 with children and adults. She was president of MOIFEM, the Québecsection of IOWME, from 1990 to 1992.

Louise Lafortune is a professor and researcher at the College André Laurendeauand a researcher at the Centre Interdisciplinaire de Recherche sur l'Apprentissageet le Developpement en Education (CIRADE) at the University of Quebec inMontreal. For her masters degree, she focused on the history of womenmathematicians and for her doctorate, on the affective domain in mathematicseducation. Her fields of interest are the role of affectivity and metacognition in thelearning process, equity in pedagoby, and qualitative research methods. She wasfounding president of MOIFEM from 1985 to 1990.

Gilah Leder is a professor of mathematics education in the Graduate School ofEducation at La Trobe University in Melbourne, Australia. Her teaching andresearch interests include gender issues, factors which affect trathematics learning,exceptionality and assessment in mathematics. She serves on various editorialboards and educational and scientific committees and is currently president of theAustralian Mathematical Sciences Council and of the Australian subcommittee ofthe International Commission en Mathematical Instruction.

Charlene Morrow is co-director of the SummerMath Program at MountHolyoke College and a lecturer in psychology and education. She is a formerpresident of Women and Mathematics Education, an affiliate group of the USNational Council of Teachers of Mathematics. She holds a doctorate in clinicalpsychology from Florida State University and her current research interests includethe investigation of women's approaches to learning mathematicsind culturalditThrences and similarities in mathematical attitudes among females. She is also anorigami enthusiast.

James Morrow is co-director of the SummerMath Program at Mount HolyokeCollege and a lecturer in mathematics. He is the executive director of the NCTMaffiliate, Women and Mathematics Education. His research interests includeinvestigating the effects of a constructivist mathematics programine for students,and surveying attitudes of minority students toward mathematics. He holds adoctorate in mathematics from Florida State University.

268

Notes On Contributors

Roberta Mura is a professor of mathematics education at Laval University inQuebec, where she is also a member of the multidisciplinary Feminist Researchgroup. She has a laurea in mathematics from the University of Milan and adoctorate, also in mathematics, from the University of Alberta. Between 1980 and1984 she was the Canadian national coordinator oflOWME.

Cornelia Niederdrenk-Felgner studied mathematics and physics at the Uni-versity of Tuebingen where she graduated with a doctoral degree. She thenbecame a secondary-school teacher of mathematics and physics, and still works asa co-author of mathematics school textbooks. Currently, she works at the GermanInstitute for Distance Education providing in-service training for mathematicsteachers, including coordinating the 'Girls and Computers' project.

Pat Rogers is a professor in mathematics and education at York University inToronto. She has a doctorate in mathematics from the University of London in thearea of mathematical logic and is currently Working in the arca of curriculumreform and effective mathematics pedagogy at the undergraduate level. For thepast five years, she served as founding academic director of the Centre for theSupport of Teaching at York University. She was Canadian national coo -dinatorof IOWME from 1984 to 1988 and co-org. ized, with Gabriele Kaiser, theIOWME sessions at ICME-7 in Quebec. Educated in England, she now lives inToronto with her daughter, Kate.

Nancy Shelley has spent a life-time in education. She pioneered the work onwomen and mathematics in Australia and was the founding Convenor of IOWME,holding that office from 1976-84. She has presented papers in the IOWMEsessions of ICME-4 in Berkeley that is in the published record of that Congress

and at ICMI2-5 in Adelaide.

Claudie Solar is a professor of mathematics education in the Faculty of Educationat the University of Ottawa. She holds a masters degree in mathematicaloptinnzation and a doctorate in education. Her fields of interest are feminism andeducation related to both children and adults, mathematics education andtechnology. She is currently doing research in the area of equity and grouplearning, inclusive pedagogy, and the use of calculators in elementary schools.She has been president of MOIFEM since 1992.

Neela Sukthankar has been living in Papua New Guinea for the past fifteen yearsand works in the Department of Mathematics and Computer Science, at the PapuaNew Guinea University of Technology. She has a doctorate in mathematics fromthe Tata Institute of Fundamental Resear:h, Bombay and her research interestsinclude gender and education in Papua New Guinea. She is the founder editor ofthe Papua New Guinea Journal of Mathematics, the founding organizer of thenationwide Papua New Guinea Mathematics Competitionind the nationalcoordinator for Papua New Guinean section oflOWME.

2694 1


Denisse R. Thompson holds a PhD in education from the University of Chicago.She currently has a position in mathematics education at the University of SouthFlorida with responsibility for preparing prospective teachers and updatingpractising teachers. Her research interests include curriculum development, theimpact of technology on curriculum, issues of assessment, the study of proof, andthe effective preparation of prospective mathematics teachers.

Sue Willis is a professor of mathematics education at Murdoch University inWestern Australia. Her research interests are in gender and education andinformed numeracy, the latter also from a social justice perspective. Her researchand curriculum-development interests are linked by a commitment to mathe-matics for social justice.

Marjolijn Witte is a researcher at the Centre for Innovation and CooperativeTechnology of the University of Amsterdam. She is interested in the design ofeducational environments open to all learners and is finishing her doctorate in thearea of girls and mathematics education.

270

Index

ability 12, 52-3, 56, 66, 73-4, )1, 111-12,129-32, 138, 151, 177-8, 180, 190,193, 219, 236, 240, 249-50, 255

avenge 27-8, 30, 32-4absolutism 209, 214, 216, 218-21.138-9.

247. 250-1abstracted discourse on maths 226-33abstraction of maths 143. 214, 238access

equality 207rewurces 156-7o counselling 53t .ducation 59, 188, 200to knowledge 173to labour market 120to maths 16. 184, 186, 215, 222to power 201-2, 204

at culturation 100-1achievement 7. 9. 28. 31. 41. 49 55 59 82

84- 5, 90-1, 93-5, 98-9. 101-6, 110,113-17. 119-20, 130-4, 150-1.183-4, 186-7, 190, 195. 198. 200-1.210, 214. 232

Adams. C. 190, 191aesthetics 215, 218. 220atiectivity in maths learning 66, 68-9ar y, sense of 187. 227algebra N. 32. 131, 171alienation 18, 172. 178Ande7son, B. 14. 15anxiety 6, 12. 68. 183

math 6, 131 3, 157, 189Arendt, Hannah 247. 254Aristotle 252. 253arithmetic 30, 82-3, 98, 109, 131, 137-9, 181Armstrong, J.M. 9, 53arts colleges 15 54assertiveness 52, 127

assimilation, cultural 110, 116attention of teacher 14, 33, 42, 60, 73, 201,

207attitude

change 11, 21-2, 47. 157. 170, 186, 198cultural 101, 247gendered 1, 9, 75-6, 90, 129, 206importance of 30-1influence of school on 47of Maori girls to nuths 109-12, 114social 117. 136-7, 140to maths 14, 68-9. 80, 172, 236. 239

attrition rates 28. 52. 184kustralia 3. 5, 89, 92, 117-24. 186-98. 250authority 253-4autonomy 145, 158. 165, 201. 227. 233, 240.

141awareness phase model 1 9. 11-12

balance. gender 2-3, 11-12, 49, 90. 120, 122.126-7, 153, 226-8, 233

feminism redressing 155-60Barne, Mary 55, 192. 193, 194, 195barriers to maths learning 14-15 55 60, 90,

105, 120, 123Barsky, D. et al 141, 143. 144Becker, Joanne Rossi 49, 53, 153. 163-73behaviour 11, 21, 47, 72, 74-5, 91, 103, 123.

136, 194, 200-1, 205-6, 211, 218."136 -7

influence of pnnt media on 117-20Belenkv, Mary Field 4, 5, 6. 8. 14, 16. 18. 1'),

24. 157, 163, 164, 165, 170, 176belief

personal 1, 19, 24, 60, 91, 103, 169,214

social 133, 198Benhow, C l 51, 52, 53

9- ,271

Index

Bennet," 249, 257bias, social 1, 15. 68. 118, 127, 200Billard, L. 15, 52Bloor,David 209. 211, 212board work 180. 182Brandon, Paul R. 7, 89. )4. 98-106Brandt. Norman 50, 51British Council 77, 79, 849rouwer. L.E.J. 237, 239. 240b)erk. Dorothy 18. 168. 169. 170, 176.

177Burton. Leone 8, 47. 49, 151, 154, 158,

209-22, 235

calculus 92. 192Campbell, P.B. 27. 28camps, summer 5-7. 8. 11. 29Cmada 12, 41-6. 66-70, 89. 117-24capability 16, 18. 164, 177. 191. 198captions writing for exhibitions 78-9.

84care 176. 178, 184career 64, 67, 73, 130

maths based 14-15. 49. 170maths related 15, 52. 235-6options F., 22. 27. 34. 90, I 1 1. 117.

119-20. 124, 141, 143-4. 171-2.186-7

challenge 11, 16, 19-20, 24. 34-5. 169.180

Chancellor College. Malawi University147-52

child centredness 21 I. 207choice 186-9, 197 -8. 201. 203, 207, 777.

235-6, 239, 241Chung. Y.Y. 129-30. 133Clark, M. 200. 201, 204class, social 82. 227classism 196, 198Clewell, B.C. 14, 15closed/open maths 237-8Cockcroft, WH. 41. 80oeducati-m 23-4. 33-4, 55, 75, 127. 137,

141-5. 148 9. 151. 157colla'ooration 5, 9, 15, 17. 22. 24 -5, 28. 41.

55, 59, 158, 170, 172. 179. 187- 3.97, 707

f 01111611 22 57 5h. 78. 143, 198Common 1 hreods exhibition 12, 77- 8( 1ommuntration :5, 19, 22-3, 29. 61,

86. 127. 179, 210, 217, 240ci )mparati es 138- 0ompetence 2. 7. 72 3, 118. 164, 184, 186,

191

272

competition 9, 15, 136. 141. 143-5. 158,170, 178, 182, 184 204-5. 207.209-10, 261

compliance. school 103-6compulsory education 4, 59. 89, 92, 95. 112.

119, 133computers, girls and 14. 72-6confidence 1, 12. 16, 19-20. 22, 27. 31-2.

35. 44-5, 52-4, 68-9, 73, 84, 91 27,

130. 177-81. 186-7, 189, 193, 23:).251

Confrey." 219, 238connected

knowing 166-8, 171. 177learning 11. 16, 18teaching 153. 168-70, 172

connection 43-6, 79-80women with maths 13-25

constraints 188, 198, 201, 222on girls action in maths 235-42

constructed knowing 165. 167-8, 170. 172constructivism. social 6, 8, 11, 16, 44. 79.

168, 210, 214, 220, 222. 226. 236,239-40, 242

contextuahzaticm 80-1, 117-18, 126, 138-9,143. 176, 190. 192, 211. 213-14. 217,229. 231-3. 236-7. 239

control 19. 21-2, 131-3. 193counselling 12, 14, 53-4, 56Countryman, Joan 2. 3, 4. 9. 37. 44. 46. 47creativity 220, 222, 240-1critical mass 12. 54. 56cross curricular activity 24. 77, 84culture

and achievement 110dependency 6-7. 211and gender 1-2. 4. 14. 82-4, 86. 89-90gender Mequity and 126- 8gender and niaths 91-6mid maths 117-24. 129. 247, 259- 61

till riculumembroidery in 85National 82. 86. 210problem based 17recorin 1-3. 9, 23, 64, 77, 127. 153. 171.

1 86. 188 90, 192, 198. 709 -10, 727,236

so, hilly ruwal 194 8osategies to benefit girls 20o 7

Dalen, D. van 239, 740Da min. Suzanne K. 6, 120, 121, 170. 210.

219. 221, 222Davis, R.B. et al 168, 22o

degreesdoctorates 49-54, 60. 63, 92, 95graduate 12, 15, 27. 54-6, 82. 90. 148,

171

masters 50. 53Delon, Francoise 7, 90, 141-5Descarries-Bélanger. F. 155, 156Desforges. C. 41, 45developed countries 92, 92-4. 136, 137. 150developing countries 89-90. 92-3, 94. 96.

126-8. 147-8development

female 77human 126. 127maths 212-13. 253-4. 259moral 153. 163-4, 176personal 28-9professional 67

disadvantage 11. 92, 120. 123, 139. 142,154-5, 163, 183. 186-8, 195, 200.206, 235

discipline. maths as 155. 158-9discomfort 54, 68discontinuity 110, 122oiscussion 16-18, 39. 42-3, 45. 61-3doing mathematics 47, 66-70, 220. 228dominance, male 2, 15, 23-4, 73. 82, 84. 94,

120, 122, 133, 138, 142. 158, 178.187,203,207,214,245,255-7.259-61

dommanon 249Dowhng, Paul 188, 195drop outs 34, 52, 154, 236dynanncs, gender 201-1 206-7

Eccles, J.S. 14, 15, 16. 23, 117. 118, 120, 113Ecoles Normales Sup&ieures (ENS) 141-5Ehot, T.S. 245, 247, 248, 249. 251, 252. 253,

256, 257. 258, 259, 260. 261ehte 141-3emotion 8, 68, 70. 158empowerment 61, 90, 158, 180-1

dis- 6. 176, 177encouragement 14, 20, 27, 29 15 53. 123.

137, 178. 180. 186-7. 207, 237environment:

gendered 55impersonal 53influence of school 41, 47learning 113, 116, 142, 145. 184. 201,

207, 219, 236-7, 242social 54

environmental destruc non 248. 257, 2o1epistemology of maths 9

Index

EQUALS project 5, 114Ernest, P 82, 209, 226, 242Ethington. C.A. 129, 133Eurocentrism 2, 213exclusion 4. 14, 68. 73, 85, 121. 164, 188expectation

family 94. 122, 202gendered 22, 129. 136. 206limited 27-8Maori 110-11of others 47sc;cial 1. 24. 73, 91, 117, 1 13-4. 188. 217,

240student 75teacher 30. 35, 74. 151. 204

experienceeducational 18. 31. 72. 80female 3, 8. 14, 24. 27-8. 56. 70, 121,

154, 163gendered 200learning 40. 154. 200mathematical 11-12, 16, 192-3personal 22, 75. 154, 166. 169, 172, 176.

178. 210, 214-17, 221, 226-9, 231,233. 236. 240. 249, 251, 254

social 256workshop 17

expert/user maths 237-42exploitation 85. i 20-1, 156, 261exploration, maths as 16, 55. 153. 190. 192.

219, 239, 241

ficilnies 28. 90, 126, 135faculty women on 12, 15, 49-50. 54. 60failure 4, 53. 69. 73, 132. 151, 175-6, 182-3,

129.152Faludi, S. 118-19Faulstich-Wieland, H. 72. 73fear 6. 43, 68, 75. 226, 252Femgold, A. 91. 93'female' maths 154, 158fenuninity 73, 85-6, 121. 189-92. 236feminism 1-3. 6. 8-9. 11-12, 16-18, (41, 66,

70, 90, 117, 119-23, 153, 163, 186-9,194-5, 197. 227

epistemolog of nuths 2(19-22redressing gender imbalance 15::--60and teaching 178-83

Fennema, Elizabeth 9, 14, 15, 49, 53, 60, 91,92, 98, 170, 201, 209

milo achievement 99, 101 -2. 105-6financial support 52. 92, 136Fontenay 142- 3Forbes, Sharleen D. 7, 89, 109-16

4.) 27.3

Index

France 90, 141-5Frazier-Kouassi, S. a al 53, 54, 56Freire. P. 159, 170, 259, 260Friedman, Lynn 12. 49-56, 91, 93. 98Fromm, E. 250, 254Fullerton. Olive 12. 37-47

games 154, 200-7gender

culture and maths 91-6ethnicity and performance 113-14and maths 1-9and maths in Papua Iew CZ:mica 135-40and maths in Singapore 129-34non-Western perspective on educational

inequity 126-8reform through school maths 186-98

geometry 30, 79. 83, 131, 138, 238George. S. 249. 257Germany 3, 72-6gifted. mathematically 51-2, 70Gilligan, Carol 8, 153, 163, 164, 167, 176Girls and Computers project 72-6Glaserfeld. E. von 16, 218. 242globality 218-19, 221goal setting 179, 201. 203. 205-6, 219Grant, Vance 50, 51graphing 131. 193Grevholm. Barhro 12. 59-64group work 180. 181, 195

Saleha Naghnn 90, 126-8Hadamard, J. 217, 218Halliday% M.A.K. 39. 42Hammond. O.W. 94, 99Hanna, G. 7, 61. 93, 94, 129Harding. Sandra 209, 211, 212, 215, 236Harris, Mary 12. 77-86, 196Harrison, J. 53. 54Haug. Frigga 227, 228, 230Hawai'i 7. 89, 94, 98-106Hekman. Susan 219. 221Hender,on, David 7, 9heteronomy 231-3Hiddleston, Pat 7, 90, 147-52hierarchy 85, 156. 158Higa. Terry Ann 7, 89, 98-106Higgins, Joanna 154. 200 -7history

absence of Wolnen in maths 15, 68, 70maths 211-13

llolland,J, 82. 85, I%lloyles, C. 41. 241Hyde, I.S. 14, 49, 91, 93, 04, 98, 99

274 9FF.

Hypatia 4. 68

identity 103, 188, 227, 248Imbert. F. 143, 144inunigration. influence on performance in

Hawai'i 98--106independence 22. 30. 181-2India 94-5, 95inferiority 137, 172. 239inquiry 209-10. 214, 218insight 216. 218, 220-1. 253inspiration 61, 63intelligence 68, 82. 188interaction

children 200-1, 203-7classroom 42cross generational 104gender 155. 157. 194. 250maths/society 154social 54, 68, 211teacher/student 14. 19, 74-5. 104

interest 5. 32, 72-3. 171, 236, 242International Organization for Women and

Maths Education (IOWME) 1. 255intervention projects 1. 5-6. 11. 12. 56, 60-1,

89, 114. 154-5, 157. 159, 187-8, 195,23;, -7

intuition 153. 166, 169, 176-7. 192, 217-18.220-1, 253

investigation 180. 192invisibility of women 60-1, 63isolation 19, 22, 54. 187, 232, 250. 255

Jackson, A. 15. 54Jacobs, J.E. 118. 120, 121. 123Jakucyn, Natalie 29. 30Johnson. R. 110. 186Johnston, Betty 8. 154, 226-33Jordan, Cathie 7. 89, 98-100Joseph, George G. 212. 213

Kaeley, Gurchan Singh 89. 91-6Kaiser. Gabriele 1-9Kaiser-Messmer, G. 3, 72. 155, 158. 159Maur, Bel inderjeet 90. 129-34KayRr, HéRne 12, 66-70Keller. Evelyn Fox 209. 214, 236Kelly, A. 209. 210Kenway. J. 186, 187, 188. 196Kimball. M.M. 59, 91kinship networks 104-5knowing

maths 210-11, 214-16, 218, 220-2way 01153, 156, 192. 249 -50, 252women's ways of 163-73

Kovalevskaja, Sonja 4, 60, 68Kundiger, E. 93, 94. 129

labourforce 27, 118. 138-9gendered 127market 64, 120, 187, 237sexual division of 73, 85. 156, 197.

230undervaluation of women 12, 296

Latbrtune, Louise 12, 66-70Lakatos. I. 209, 238. 240Lamon, S.J. 14, 49, 91, 98Lange, J. de 237. 241language 4, 12. 90, 156, 210

academic 175, 177. 181, 184, 191as bar:ier 110, 131, 135. 138-9maths 154. 237-42

Larouche, C. 93. 94. 129leadership 11. 28, 33learning

affectivity in maths 66, 68-9development 30evaluation 182-3maths 194prekrence 8style 7, 12. 24, 157. 183, 205-7. 209.

221...oring 175. 179. 183-4

Leder, Gbh 14. 89, 90, 91, 92. 117-24, 170,209. 235

Lee, V.E. 55. 148legislation 55. 121-2Lerman, S. 238, 240Leuar. B.C. 129, 130. 133Lewis. D.J. 52, 56listening 41, 166, 179-80logic 167. 171. 176, 253-5

McClintock. Barbara 214. 256McIntosh, Peggy 1. 2. 3. 4. 6. 9. 11, 12Malawi 90, 147-52Maori 7, 89, 109-16marginalization 23. 85mathematician:

being a 216-20. 259image of 67-8

Maths in Work projet t 78 9, 81, 83matrilmy 89, 95measurement 109. 131, 229, 232memorization 16, 23, 1911-1memory-work 227-8Meredith. Chat !es W 30 -1, 1(11Meredith. G.M. 100. 101

METRO achievement programme 11.27-35

modelling 45. 55, 56, 192, 237. 241monoculturalism 2-3, 5, 9. 11, 89-90Morrow Charlene 6, 11, 13-25, 155. 157.

168. 169Morrow. James 6, 11, 13-25, 155, 157motivation 22. 27, 61. 91. 119, 114. 186, 205.

237. 241Mura, Roberta 153, 155-60, 256, 260mystification 197-8

de- 66. 69, 178, 214, 221

narrative 213-14, 218, 221needlework and maths 77-86, 190. 196Nelson, D. et al 212, 213Netherlands 235-42network 12, 66-7. 142

of Swedish women 59-64New Zealand 7, 89, 109-16Newton, B.J. 94. 99Newton, Isaac 250, 151Niederdrenk-FeIgner, Cornelia 12. 72-6Noether. Emmy 4, 15, 68, 145number '8-9. 83. 91. 118, 226

objectivizy 14. 176. 178. 209-15. 217-19.227, 237-40, 254-5

observation 215, 221one-minute papers 181-2opportunity 11. 31, 119. 123. 136-7. 188.

190, 202-3equality 46. 120, 139. 207

oppression 6. 113, 120-1, .56-8. 197, 249.256. 261

ordering 239, 241otherness 252-3ownership 178. 184. 191, 201, 207. 256

Papua New Guinea 90, 93, 95, 135-4(1peace and war 257-61peers 27 Si 54 89, 98-1115Penrose. Roger 216. 217, 218perception 195. 222. 236. 253performance 14, 59, 73, 89, 123. 130-3, 137.

149-52, 170. 172. 191. 210. 212, 141gender and culture 91-6Maori and white New Zealander 109,

112 -15personalization 214. :26- 18. 221Poinear, R.N. 217, 218Polanyi, M. 210, 251potential 27. 3J-5, 73, 91, 126, 136poverty 90, 12.t. 187. 248

9I

275

1

ludo

power 8, 20, 46. 86. 92. 95-6. 100. 124.141-2, 153, 155, 158, 166. 188. 192.194, 201-2, 204. 213. 220. 232.248-50, 252, 254. 256. 259-61

Priest, M. 138primary education 41, 92-3, 95, 103, 135,

147, 149. 200elementary school 14, 43, 53, 99, 104. 154

print media 89-90, 117-19prohlcm solving 13, 15-20. 22, 28, 55. 69.

91, 130-1. 133, 138. 168-9. 171, 175,177. 180, 207. 219. 238-41

procedural knowing 165-7, 172proof gimeration 181-2, 212-13. 253proportion 80. 83psychology 90. 158

questioning 16. 19-21. 166. 170. 179-80,182

questionnaires 31-2, 79-80, 109-11. 129-30

race 14-15, 22, 28-30, 34. 82. 84, 94-5, 132,227, 248

comparison of Maori and white mathsperformance 1(19-16

minority achievement in Hawai'l 99,101-3. 105

racism 105. 123. 187. 196ratio, mak/female and maths 15. 49-53. 60.

92reading maths 30. 179-82. 195reality 30. 43-6, 69-70, 81-3. 153-4,

189.-90. 193. 229-31. 233, 237-9,247. 249, 254, 256, 258. 260-1

reasoning 8. 22, 56. 167. 176, 232. 241, 253received knowing 165-6register. maths 42-3. 43-6relevance of maths 43-4, 46, 111-12, 115.

130, 190, 236, 239representation of women in maths 6. 15. 49,

68, 172resean. h 1. 8-9. 14-15. 41. 52, 72. 74,77,

92-3, 98. 129-34resource inaterials Common Thieads

development 79, 81- 3development 64disruptive 194 7gender equality 190girls and L oniputers 74lal k of 28METRO maths 30utilization 127

responsibility 176. 178, 193, 221. 252Restwo, Sal 213, 215

276

risk taking 11, 19. 181. 186, 204, 207rivalry 104, 248Rogers, Pat 1-9, 153, 155. 158, 159, 175-84role

child assuming teacher 203-4, 206education 135ENS 141-2, 145t:ither 100gendered 23-4. 90, 117. 120. 2(14. 227maths 137, 159. 198models 15. 29, 137, 144. 186scientist 17teacher 19. 67. 82. 175teacher/student 170women 92, 93, 95, 122, 136

Rose. Hilary 209. 221Rosser. Susan V. 209. 210, 211. 215. 221Roy. S. 155. 156rule-following 190-1

Sadker, M. 14. 42Saint-Cloud 142-3Sampson, S.N. 118, 209Schifter, D. 19, 23science 209-22secondary education 59-60, 92-3. 99.

109-15, 127. 133. 135, 137, 147, 226.236-7

high school 14-15. 27. 29, 31-4. 43, 51.91, 93. 98. 101, 149

segregation 61. 75, 85. 127. 137. 155. 157.159

de- 90, 141-5. 187self 5. 18, 31, 132. 251

-confirmation 18-19, 169-esteem 16. 47. 69. 186-7. 189

separate knowing 166-8. 171, 177,Scsanw Street 118

Siwres 142-3sexism 6. 144. 153. 156, 158-9, 186, 189-90,

192, 194-0. 198sexuality 156. 227Seymour, E 51 56Sharpe. N.R. 15. 54Shelley, Nancy 9. 245, 247-61Sherman.). 53, 235silence 165. ' 70-8. 229, 250

Q.E. 129. 132Snn. WK. 131, 133SIMS (Second International Mathcinams

Study) survey 7, 109-16Singapore 90. 129-34single sex education 7. 22-4, 28 11 55

911. 137. 147-52

skill 6. 11. 22. 29, 43. 5L, )5. 73-4, 79-82.83. 98. 179-81. 187. 196-7, 207, 247

cognitive 55, 91. 93, 95, 131. 164, 177,201

verbal 99. 102. 106Smith. D. and R. 258, 239Smith. Dorothy 226. 227social-psychological approach to culture

influence 117, 119-20sociaLzation 91. 120. 156-7. 214. 221. 227socirny 8. 121, 140. 237

economic CR.-tors and 126-8traditional 93-4, 135-7, 139-40

Solar, Claudie 12, 66-70space time 247. 250-2spatial visualization 20. 70, 91. 131. 138-9.

236speed 69. 91Stinky. J. 51, 52status

socio-economic 28, 1(19- 13. 115. 132.136. 142-3. 187-8

of women in maths 4. 66-7stereotyping 24. 27. 53, 67. 73-5, 85-6, 91,

118. 123-4. 130. 132, 136, 144-5.159. 164, 196

strategies to benefit girls ui th 1

100-7.__e c_assroom

Streitmatter, J. 91. 92student sensitive teaching 178-82subjective knowing 165-6subjectivity 121. 176, 171% 211. 218. 227,

139-40subordniation 14, 85. 136, 177. 203-4. 227success 4. 15, 28, 30, 44-5, 94, 101 11(', 11(1,

130. 141. 144, 149. 151. 187, 19o- 1,207, 214

imagery 119. 121- 3Sukthaiikar. Neela 133-4oSunmierMath 6, 11. 13-23. 17-10superiority 91. 94. 131. 159support 11. 16 -17, 19-20. 22. 24. 28. 49,

33-5. 39. 61, 93, 161 18o, 186Sweden 3, 12, 59-64symbolism. mathemant 09 70, 240symnwtry 78- 9. 191)

talking maths 39 -42Ian. 0 S. 129. 131 2. 133Tan/er. N.K. 129. 132teat hmg

approat hes to women 66. 70and fenmusin 153, 158 9maths in theory and prat ut e 175 84

Index

narrowness 84reform 8. 134, 236style 172

iechnology 5. 12, 29, 30. 73. 83. 138testing 18. 59-60. 91, 99. 132. 141. 143-3.

172. 173. 182-3. 241GCE '0' le%el 129-31. 133Malawi Certificate of Education 147-9SAT-M 31-2Schools Certificate m New Zealand

114-15textbook 4. 61. 63-4, 75. 81. 84. 177. 189.

192-3. 195-7. 210. 238-9think-write-pair-discussion 179-81Thom, R. 217. 218Thompson. Demsse R. 11. 27-35time, concept of 138-9. 229Tobias, Sheila 6, 119, 155, 157, 172training, teacher 12. 37-47. 59. 64. 72, 74,

142transmission. teaching as 173-7. 183, 209.

217. 219-20, 235,138trust 22, 105, 181-2truth 166. 168, 170-1, 210. 212. 216-18.

140typology of personal and curricular revision

1--9

Ulm 142-3understanding 14. 16-17. 19-20. 22-3. 47.

66. 80-1, 131. 133, 139, 171 177.179, 181-2. 184. 191-3. 210, 213,217, 219-20. 231. 233. 247.149. 231

United Kingdom 3. 5. 92. 94. 100. 210United States of America 1. 3-5, 14. 27-8.

32. 49. 31 55-6. 91--4. 98. 100-1umversity:

French 90Malawi 147-52Papua New Guinea 95researt h 15teak her trannng in 37-47women in nuths m

valueof education 101free 20. 211. 2'0, 238laden 26(1of maths 15. 21. 56, 08. 90. 111. 115. 124.

130, 137system 164. 201

Verhage. II eleen 189, 19o, 195, 1%violence 139 40, 136, 248. 254. 260- 1vok e 8. 20. 22, 55, 164. 169, 178, 184

.....

277

Index

Walden, R. 191, 203, 204Walkerdine, Valerie 47, 85, 188, 190. 191,

192, 203, 204, 226, 231, 232, 236Willis. Sue 86, 153, 155, 158, 159, 186-98,

227Witte, Marjolijn 8. 154, 235-42women

as central to maths 8-9, 11-less maths 4in maths 4-5.15 problem in maths 5-6, 11-12

Women Do Math 12

2782 ci

Women and Mathematics 59-64, 67-70Wnnen's Ways qf Knowing: The Developtnent tf

Self, Voke and Mind (Belenky et al) 163,164. 165, 168, 171, 172, 173

workplace, maths in 12, 79-83, 122workshops, maths 17, 66-70. 186

Yip, S.K.J. 131. 133Young. Grace Chisholm 68

Zani, J. 251, 261

Liar

Equity in Mathematics Education:Influences of Feminism and Culture

Edited byPat Rogers and Gabriele Kaiser

Research and intervention over the past three decades have greatlyincreased our understanding of the relationship between gender andparticipation in mathematics education. Research, most of it

quantitative, has taught us that gender differences in mathematicsachievement and participation are not due to biology, but to complexinteractions among social and cuttural factors, societal expectations,personal belief systems and confidence levels. Intervention to alterthe impact of these interactions has proved successful, at least in theshort term. Typically, interventions sought to remedy perceived 'de-ficits' in women's attitudes and/or aptitudes in mathematics by meansof 'special programmes' and 'experimental treatments'. But recentadvances in scholarship regarding the teaching and learning efmathematics have brought new insights. .Current research, profoundlyinfluenced by feminist thought and methods of enquiry, hasestablished how a fuller understanding of the nature of mathematics asa discipline, and different, more inclusive instructional practices canremove traditional obstacles that have thwarted the success of womeilin this important field. Some argue that practices arising out ofcontemporary analysis will improve the study of mathematics for allstudents, male and female afte. This book provides teachers,educators and other interested readers with an overview of the mostrecent developments and changes in the field of gender andmathematics education. Many of the chapters in this volume arose outof sessions.

Pat Rogers is an Associate Professor of Mathematics and Education at YorkUniversity, Canada. She is well known in North America for her activist workin advancing the issue of women and mathematics education, and is currentlypromoting feminist teaching methods and curriculum transformation in

mathematics. In recognition of her work by being appointed to the GeorgePolya Lectureship of the Mathematics Association of America for 1992-94Gabriele Kaiser has worked as a researcher in the Department ofMathematics and Computer Science Studies at Kassel University, Germany,since 1981. Besides her continuing engagement in women and mathematicseducation, she is currently working in the area of comparative empiricalresearch concerning the teaching of mathematics in context with particularreference to England and Germany

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DOCUMENT RESUME ED 391 849 UD 030 791 AUTHOR TITLE 95 … · DOCUMENT RESUME. ED 391 849. UD 030 791. AUTHOR Rogers, Pat, Ed.; Kaiser, Gabriele, Ed. TITLE Equity in Mathematics Education